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Theorem dfrdg4 35786
Description: A quantifier-free definition of the recursive definition generator. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
Assertion
Ref Expression
dfrdg4 rec(𝐹, 𝐴) = (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))))

Proof of Theorem dfrdg4
Dummy variables 𝑎 𝑏 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrdg3 35631 . 2 rec(𝐹, 𝐴) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))}
2 an12 643 . . . . . . . 8 ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))) ↔ (𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
3 df-fn 6547 . . . . . . . . . 10 (𝑓 Fn 𝑥 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝑥))
4 ancom 459 . . . . . . . . . 10 ((Fun 𝑓 ∧ dom 𝑓 = 𝑥) ↔ (dom 𝑓 = 𝑥 ∧ Fun 𝑓))
5 eqcom 2733 . . . . . . . . . . 11 (dom 𝑓 = 𝑥𝑥 = dom 𝑓)
65anbi1i 622 . . . . . . . . . 10 ((dom 𝑓 = 𝑥 ∧ Fun 𝑓) ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
73, 4, 63bitri 296 . . . . . . . . 9 (𝑓 Fn 𝑥 ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
87anbi1i 622 . . . . . . . 8 ((𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))) ↔ ((𝑥 = dom 𝑓 ∧ Fun 𝑓) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
9 anass 467 . . . . . . . 8 (((𝑥 = dom 𝑓 ∧ Fun 𝑓) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))) ↔ (𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))))
102, 8, 93bitri 296 . . . . . . 7 ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))) ↔ (𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))))
1110exbii 1843 . . . . . 6 (∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))))
12 vex 3467 . . . . . . . 8 𝑓 ∈ V
1312dmex 7912 . . . . . . 7 dom 𝑓 ∈ V
14 eleq1 2814 . . . . . . . . 9 (𝑥 = dom 𝑓 → (𝑥 ∈ On ↔ dom 𝑓 ∈ On))
15 raleq 3312 . . . . . . . . 9 (𝑥 = dom 𝑓 → (∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ↔ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
1614, 15anbi12d 630 . . . . . . . 8 (𝑥 = dom 𝑓 → ((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))) ↔ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
1716anbi2d 628 . . . . . . 7 (𝑥 = dom 𝑓 → ((Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))))
1813, 17ceqsexv 3516 . . . . . 6 (∃𝑥(𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
1911, 18bitri 274 . . . . 5 (∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
20 df-rex 3061 . . . . 5 (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
21 eldif 3957 . . . . . 6 (𝑓 ∈ (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ (𝑓 ∈ ( Funs ∩ (Domain “ On)) ∧ ¬ 𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))))
22 elin 3963 . . . . . . . 8 (𝑓 ∈ ( Funs ∩ (Domain “ On)) ↔ (𝑓 Funs 𝑓 ∈ (Domain “ On)))
2312elfuns 35750 . . . . . . . . 9 (𝑓 Funs ↔ Fun 𝑓)
2412elima 6065 . . . . . . . . . 10 (𝑓 ∈ (Domain “ On) ↔ ∃𝑥 ∈ On 𝑥Domain𝑓)
25 df-rex 3061 . . . . . . . . . 10 (∃𝑥 ∈ On 𝑥Domain𝑓 ↔ ∃𝑥(𝑥 ∈ On ∧ 𝑥Domain𝑓))
26 vex 3467 . . . . . . . . . . . . . . 15 𝑥 ∈ V
2726, 12brcnv 5880 . . . . . . . . . . . . . 14 (𝑥Domain𝑓𝑓Domain𝑥)
2812, 26brdomain 35768 . . . . . . . . . . . . . 14 (𝑓Domain𝑥𝑥 = dom 𝑓)
2927, 28bitri 274 . . . . . . . . . . . . 13 (𝑥Domain𝑓𝑥 = dom 𝑓)
3029anbi1ci 624 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑥Domain𝑓) ↔ (𝑥 = dom 𝑓𝑥 ∈ On))
3130exbii 1843 . . . . . . . . . . 11 (∃𝑥(𝑥 ∈ On ∧ 𝑥Domain𝑓) ↔ ∃𝑥(𝑥 = dom 𝑓𝑥 ∈ On))
3213, 14ceqsexv 3516 . . . . . . . . . . 11 (∃𝑥(𝑥 = dom 𝑓𝑥 ∈ On) ↔ dom 𝑓 ∈ On)
3331, 32bitri 274 . . . . . . . . . 10 (∃𝑥(𝑥 ∈ On ∧ 𝑥Domain𝑓) ↔ dom 𝑓 ∈ On)
3424, 25, 333bitri 296 . . . . . . . . 9 (𝑓 ∈ (Domain “ On) ↔ dom 𝑓 ∈ On)
3523, 34anbi12i 626 . . . . . . . 8 ((𝑓 Funs 𝑓 ∈ (Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
3622, 35bitri 274 . . . . . . 7 (𝑓 ∈ ( Funs ∩ (Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
3736anbi1i 622 . . . . . 6 ((𝑓 ∈ ( Funs ∩ (Domain “ On)) ∧ ¬ 𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ¬ 𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))))
38 brdif 5197 . . . . . . . . . . . . . . 15 (𝑓(( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦 ↔ (𝑓( E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦))
39 vex 3467 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
4012, 39brco 5868 . . . . . . . . . . . . . . . . 17 (𝑓( E ∘ Domain)𝑦 ↔ ∃𝑥(𝑓Domain𝑥𝑥 E 𝑦))
4128anbi1i 622 . . . . . . . . . . . . . . . . . . 19 ((𝑓Domain𝑥𝑥 E 𝑦) ↔ (𝑥 = dom 𝑓𝑥 E 𝑦))
4241exbii 1843 . . . . . . . . . . . . . . . . . 18 (∃𝑥(𝑓Domain𝑥𝑥 E 𝑦) ↔ ∃𝑥(𝑥 = dom 𝑓𝑥 E 𝑦))
43 breq1 5147 . . . . . . . . . . . . . . . . . . 19 (𝑥 = dom 𝑓 → (𝑥 E 𝑦 ↔ dom 𝑓 E 𝑦))
4413, 43ceqsexv 3516 . . . . . . . . . . . . . . . . . 18 (∃𝑥(𝑥 = dom 𝑓𝑥 E 𝑦) ↔ dom 𝑓 E 𝑦)
4542, 44bitri 274 . . . . . . . . . . . . . . . . 17 (∃𝑥(𝑓Domain𝑥𝑥 E 𝑦) ↔ dom 𝑓 E 𝑦)
4613, 39brcnv 5880 . . . . . . . . . . . . . . . . . 18 (dom 𝑓 E 𝑦𝑦 E dom 𝑓)
4713epeli 5579 . . . . . . . . . . . . . . . . . 18 (𝑦 E dom 𝑓𝑦 ∈ dom 𝑓)
4846, 47bitri 274 . . . . . . . . . . . . . . . . 17 (dom 𝑓 E 𝑦𝑦 ∈ dom 𝑓)
4940, 45, 483bitri 296 . . . . . . . . . . . . . . . 16 (𝑓( E ∘ Domain)𝑦𝑦 ∈ dom 𝑓)
5049anbi1i 622 . . . . . . . . . . . . . . 15 ((𝑓( E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦) ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ 𝑓 Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦))
5138, 50bitri 274 . . . . . . . . . . . . . 14 (𝑓(( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦 ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ 𝑓 Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦))
52 onelon 6391 . . . . . . . . . . . . . . . . . . . . . . . 24 ((dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → 𝑦 ∈ On)
53523adant1 1127 . . . . . . . . . . . . . . . . . . . . . . 23 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → 𝑦 ∈ On)
54 brun 5195 . . . . . . . . . . . . . . . . . . . . . . . . 25 (⟨𝑓, 𝑦⟩(((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ↔ (⟨𝑓, 𝑦⟩((V × {∅}) × { {𝐴}})𝑥 ∨ ⟨𝑓, 𝑦⟩((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))𝑥))
55 brxp 5722 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (⟨𝑓, 𝑦⟩((V × {∅}) × { {𝐴}})𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ∧ 𝑥 ∈ { {𝐴}}))
56 opelxp 5709 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ {∅}))
5712, 56mpbiran 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ 𝑦 ∈ {∅})
58 velsn 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
5957, 58bitri 274 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ 𝑦 = ∅)
60 velsn 4640 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 ∈ { {𝐴}} ↔ 𝑥 = {𝐴})
6159, 60anbi12i 626 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ∧ 𝑥 ∈ { {𝐴}}) ↔ (𝑦 = ∅ ∧ 𝑥 = {𝐴}))
6255, 61bitri 274 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑓, 𝑦⟩((V × {∅}) × { {𝐴}})𝑥 ↔ (𝑦 = ∅ ∧ 𝑥 = {𝐴}))
63 brun 5195 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (⟨𝑓, 𝑦⟩((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))𝑥 ↔ (⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ∨ ⟨𝑓, 𝑦⟩((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))𝑥))
6426brresi 5989 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ∧ ⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥))
65 opelxp 5709 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ (𝑓 ∈ V ∧ 𝑦 Limits ))
6612, 65mpbiran 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ 𝑦 Limits )
6739ellimits 35745 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 Limits ↔ Lim 𝑦)
6866, 67bitri 274 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ Lim 𝑦)
69 opex 5461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑓, 𝑦⟩ ∈ V
7069, 26brco 5868 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥 ↔ ∃𝑧(⟨𝑓, 𝑦⟩Img𝑧𝑧 Bigcup 𝑥))
71 vex 3467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑧 ∈ V
7212, 39, 71brimg 35772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (⟨𝑓, 𝑦⟩Img𝑧𝑧 = (𝑓𝑦))
7326brbigcup 35733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 Bigcup 𝑥 𝑧 = 𝑥)
7472, 73anbi12i 626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((⟨𝑓, 𝑦⟩Img𝑧𝑧 Bigcup 𝑥) ↔ (𝑧 = (𝑓𝑦) ∧ 𝑧 = 𝑥))
7574exbii 1843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑧(⟨𝑓, 𝑦⟩Img𝑧𝑧 Bigcup 𝑥) ↔ ∃𝑧(𝑧 = (𝑓𝑦) ∧ 𝑧 = 𝑥))
7612imaex 7917 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓𝑦) ∈ V
77 unieq 4917 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 = (𝑓𝑦) → 𝑧 = (𝑓𝑦))
7877eqeq1d 2728 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = (𝑓𝑦) → ( 𝑧 = 𝑥 (𝑓𝑦) = 𝑥))
7976, 78ceqsexv 3516 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∃𝑧(𝑧 = (𝑓𝑦) ∧ 𝑧 = 𝑥) ↔ (𝑓𝑦) = 𝑥)
80 eqcom 2733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ( (𝑓𝑦) = 𝑥𝑥 = (𝑓𝑦))
8179, 80bitri 274 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑧(𝑧 = (𝑓𝑦) ∧ 𝑧 = 𝑥) ↔ 𝑥 = (𝑓𝑦))
8270, 75, 813bitri 296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥𝑥 = (𝑓𝑦))
8368, 82anbi12i 626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ∧ ⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥) ↔ (Lim 𝑦𝑥 = (𝑓𝑦)))
8464, 83bitri 274 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ↔ (Lim 𝑦𝑥 = (𝑓𝑦)))
8526brresi 5989 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (⟨𝑓, 𝑦⟩((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥))
86 opelxp 5709 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ ran Succ))
8712, 86mpbiran 707 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ 𝑦 ∈ ran Succ)
8839elrn 5891 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ ran Succ ↔ ∃𝑧 𝑧Succ𝑦)
8971, 39brsuccf 35776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧Succ𝑦𝑦 = suc 𝑧)
9089exbii 1843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑧 𝑧Succ𝑦 ↔ ∃𝑧 𝑦 = suc 𝑧)
9187, 88, 903bitri 296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ ∃𝑧 𝑦 = suc 𝑧)
9269, 26brco 5868 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥 ↔ ∃𝑎(⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎𝑎FullFun𝐹𝑥))
93 vex 3467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 𝑎 ∈ V
9469, 93brco 5868 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ↔ ∃𝑧(⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧𝑧Apply𝑎))
9512, 39, 71brpprod3a 35721 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ↔ ∃𝑎𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎𝑦 Bigcup 𝑏))
96 3anrot 1097 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎𝑦 Bigcup 𝑏) ↔ (𝑓 I 𝑎𝑦 Bigcup 𝑏𝑧 = ⟨𝑎, 𝑏⟩))
9793ideq 5850 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑓 I 𝑎𝑓 = 𝑎)
98 equcom 2014 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑓 = 𝑎𝑎 = 𝑓)
9997, 98bitri 274 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑓 I 𝑎𝑎 = 𝑓)
100 vex 3467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 𝑏 ∈ V
101100brbigcup 35733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑦 Bigcup 𝑏 𝑦 = 𝑏)
102 eqcom 2733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ( 𝑦 = 𝑏𝑏 = 𝑦)
103101, 102bitri 274 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑦 Bigcup 𝑏𝑏 = 𝑦)
104 biid 260 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑎, 𝑏⟩)
10599, 103, 1043anbi123i 1152 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑓 I 𝑎𝑦 Bigcup 𝑏𝑧 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝑓𝑏 = 𝑦𝑧 = ⟨𝑎, 𝑏⟩))
10696, 105bitri 274 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎𝑦 Bigcup 𝑏) ↔ (𝑎 = 𝑓𝑏 = 𝑦𝑧 = ⟨𝑎, 𝑏⟩))
1071062exbii 1844 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (∃𝑎𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎𝑦 Bigcup 𝑏) ↔ ∃𝑎𝑏(𝑎 = 𝑓𝑏 = 𝑦𝑧 = ⟨𝑎, 𝑏⟩))
108 vuniex 7740 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 𝑦 ∈ V
109 opeq1 4872 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑎 = 𝑓 → ⟨𝑎, 𝑏⟩ = ⟨𝑓, 𝑏⟩)
110109eqeq2d 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑎 = 𝑓 → (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑓, 𝑏⟩))
111 opeq2 4873 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑏 = 𝑦 → ⟨𝑓, 𝑏⟩ = ⟨𝑓, 𝑦⟩)
112111eqeq2d 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑏 = 𝑦 → (𝑧 = ⟨𝑓, 𝑏⟩ ↔ 𝑧 = ⟨𝑓, 𝑦⟩))
11312, 108, 110, 112ceqsex2v 3522 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (∃𝑎𝑏(𝑎 = 𝑓𝑏 = 𝑦𝑧 = ⟨𝑎, 𝑏⟩) ↔ 𝑧 = ⟨𝑓, 𝑦⟩)
11495, 107, 1133bitri 296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧𝑧 = ⟨𝑓, 𝑦⟩)
115114anbi1i 622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧𝑧Apply𝑎) ↔ (𝑧 = ⟨𝑓, 𝑦⟩ ∧ 𝑧Apply𝑎))
116115exbii 1843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (∃𝑧(⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧𝑧Apply𝑎) ↔ ∃𝑧(𝑧 = ⟨𝑓, 𝑦⟩ ∧ 𝑧Apply𝑎))
117 opex 5461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑓, 𝑦⟩ ∈ V
118 breq1 5147 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑧 = ⟨𝑓, 𝑦⟩ → (𝑧Apply𝑎 ↔ ⟨𝑓, 𝑦⟩Apply𝑎))
119117, 118ceqsexv 3516 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (∃𝑧(𝑧 = ⟨𝑓, 𝑦⟩ ∧ 𝑧Apply𝑎) ↔ ⟨𝑓, 𝑦⟩Apply𝑎)
12012, 108, 93brapply 35773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (⟨𝑓, 𝑦⟩Apply𝑎𝑎 = (𝑓 𝑦))
121119, 120bitri 274 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (∃𝑧(𝑧 = ⟨𝑓, 𝑦⟩ ∧ 𝑧Apply𝑎) ↔ 𝑎 = (𝑓 𝑦))
12294, 116, 1213bitri 296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎𝑎 = (𝑓 𝑦))
12393, 26brfullfun 35783 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑎FullFun𝐹𝑥𝑥 = (𝐹𝑎))
124122, 123anbi12i 626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎𝑎FullFun𝐹𝑥) ↔ (𝑎 = (𝑓 𝑦) ∧ 𝑥 = (𝐹𝑎)))
125124exbii 1843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑎(⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎𝑎FullFun𝐹𝑥) ↔ ∃𝑎(𝑎 = (𝑓 𝑦) ∧ 𝑥 = (𝐹𝑎)))
126 fvex 6904 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 𝑦) ∈ V
127 fveq2 6891 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑎 = (𝑓 𝑦) → (𝐹𝑎) = (𝐹‘(𝑓 𝑦)))
128127eqeq2d 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑎 = (𝑓 𝑦) → (𝑥 = (𝐹𝑎) ↔ 𝑥 = (𝐹‘(𝑓 𝑦))))
129126, 128ceqsexv 3516 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑎(𝑎 = (𝑓 𝑦) ∧ 𝑥 = (𝐹𝑎)) ↔ 𝑥 = (𝐹‘(𝑓 𝑦)))
13092, 125, 1293bitri 296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥𝑥 = (𝐹‘(𝑓 𝑦)))
13191, 130anbi12i 626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥) ↔ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))
13285, 131bitri 274 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑓, 𝑦⟩((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))𝑥 ↔ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))
13384, 132orbi12i 912 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ∨ ⟨𝑓, 𝑦⟩((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))𝑥) ↔ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
13463, 133bitri 274 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑓, 𝑦⟩((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))𝑥 ↔ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
13562, 134orbi12i 912 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((⟨𝑓, 𝑦⟩((V × {∅}) × { {𝐴}})𝑥 ∨ ⟨𝑓, 𝑦⟩((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))𝑥) ↔ ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
13654, 135bitri 274 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑓, 𝑦⟩(((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ↔ ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
137 onzsl 7846 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ On ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ (𝑦 ∈ V ∧ Lim 𝑦)))
138 nlim0 6425 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ¬ Lim ∅
139 limeq 6378 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ∅ → (Lim 𝑦 ↔ Lim ∅))
140138, 139mtbiri 326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → ¬ Lim 𝑦)
141140intnanrd 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → ¬ (Lim 𝑦𝑥 = (𝑓𝑦)))
142 nsuceq0 6449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 suc 𝑧 ≠ ∅
143 neeq2 2994 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = ∅ → (suc 𝑧𝑦 ↔ suc 𝑧 ≠ ∅))
144142, 143mpbiri 257 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = ∅ → suc 𝑧𝑦)
145144necomd 2986 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = ∅ → 𝑦 ≠ suc 𝑧)
146145neneqd 2935 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ∅ → ¬ 𝑦 = suc 𝑧)
147146nexdv 1932 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → ¬ ∃𝑧 𝑦 = suc 𝑧)
148147intnanrd 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → ¬ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))
149 ioran 981 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))) ↔ (¬ (Lim 𝑦𝑥 = (𝑓𝑦)) ∧ ¬ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
150141, 148, 149sylanbrc 581 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = ∅ → ¬ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
151 orel2 888 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (¬ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))) → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → (𝑦 = ∅ ∧ 𝑥 = {𝐴})))
152150, 151syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ∅ → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → (𝑦 = ∅ ∧ 𝑥 = {𝐴})))
153 iftrue 4530 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ∅ → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = if(𝐴 ∈ V, 𝐴, ∅))
154 unisnif 35760 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
155153, 154eqtr4di 2784 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = {𝐴})
156155eqeq2d 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ↔ 𝑥 = {𝐴}))
157156biimprd 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = ∅ → (𝑥 = {𝐴} → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
158157adantld 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ∅ → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
159152, 158syld 47 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = ∅ → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
160156biimpd 228 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → 𝑥 = {𝐴}))
161160anc2li 554 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → (𝑦 = ∅ ∧ 𝑥 = {𝐴})))
162 orc 865 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
163161, 162syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))))
164159, 163impbid 211 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = ∅ → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
165 neeq1 2993 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = suc 𝑧 → (𝑦 ≠ ∅ ↔ suc 𝑧 ≠ ∅))
166142, 165mpbiri 257 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = suc 𝑧𝑦 ≠ ∅)
167166neneqd 2935 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = suc 𝑧 → ¬ 𝑦 = ∅)
168167intnanrd 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = suc 𝑧 → ¬ (𝑦 = ∅ ∧ 𝑥 = {𝐴}))
169168rexlimivw 3141 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ (𝑦 = ∅ ∧ 𝑥 = {𝐴}))
170 orel1 886 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (¬ (𝑦 = ∅ ∧ 𝑥 = {𝐴}) → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
171169, 170syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
172 nlimsucg 7842 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ V → ¬ Lim suc 𝑧)
173172elv 3469 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ¬ Lim suc 𝑧
174 limeq 6378 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = suc 𝑧 → (Lim 𝑦 ↔ Lim suc 𝑧))
175173, 174mtbiri 326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = suc 𝑧 → ¬ Lim 𝑦)
176175rexlimivw 3141 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ Lim 𝑦)
177176intnanrd 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ (Lim 𝑦𝑥 = (𝑓𝑦)))
178 orel1 886 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (¬ (Lim 𝑦𝑥 = (𝑓𝑦)) → (((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))) → (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
179177, 178syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))) → (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
180142neii 2932 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ¬ suc 𝑧 = ∅
181180iffalsei 4534 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 if(suc 𝑧 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))) = if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))
182 iffalse 4533 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (¬ Lim suc 𝑧 → if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧))) = (𝐹‘(𝑓 suc 𝑧)))
18371, 172, 182mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧))) = (𝐹‘(𝑓 suc 𝑧))
184181, 183eqtri 2754 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 if(suc 𝑧 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))) = (𝐹‘(𝑓 suc 𝑧))
185 eqeq1 2730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
186 unieq 4917 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 = suc 𝑧 𝑦 = suc 𝑧)
187186fveq2d 6895 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑦 = suc 𝑧 → (𝑓 𝑦) = (𝑓 suc 𝑧))
188187fveq2d 6895 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = suc 𝑧 → (𝐹‘(𝑓 𝑦)) = (𝐹‘(𝑓 suc 𝑧)))
189174, 188ifbieq2d 4550 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = suc 𝑧 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))) = if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧))))
190185, 189ifbieq2d 4550 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = if(suc 𝑧 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))))
191184, 190, 1883eqtr4a 2792 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = (𝐹‘(𝑓 𝑦)))
192191rexlimivw 3141 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = (𝐹‘(𝑓 𝑦)))
193192eqeq2d 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ↔ 𝑥 = (𝐹‘(𝑓 𝑦))))
194193biimprd 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = (𝐹‘(𝑓 𝑦)) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
195194adantld 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ((∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
196171, 179, 1953syld 60 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
197 rexex 3066 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ∃𝑧 𝑦 = suc 𝑧)
198193biimpd 228 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → 𝑥 = (𝐹‘(𝑓 𝑦))))
199 olc 866 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))) → ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
200199olcd 872 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))) → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
201197, 198, 200syl6an 682 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))))
202196, 201impbid 211 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
203140con2i 139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (Lim 𝑦 → ¬ 𝑦 = ∅)
204203intnanrd 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Lim 𝑦 → ¬ (𝑦 = ∅ ∧ 𝑥 = {𝐴}))
205204, 170syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑦 → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
206175exlimiv 1926 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∃𝑧 𝑦 = suc 𝑧 → ¬ Lim 𝑦)
207206con2i 139 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (Lim 𝑦 → ¬ ∃𝑧 𝑦 = suc 𝑧)
208207intnanrd 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Lim 𝑦 → ¬ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))
209 orel2 888 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))) → (((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))) → (Lim 𝑦𝑥 = (𝑓𝑦))))
210208, 209syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑦 → (((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))) → (Lim 𝑦𝑥 = (𝑓𝑦))))
211203iffalsed 4535 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (Lim 𝑦 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))
212 iftrue 4530 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (Lim 𝑦 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))) = (𝑓𝑦))
213211, 212eqtrd 2766 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (Lim 𝑦 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = (𝑓𝑦))
214213eqeq2d 2737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ↔ 𝑥 = (𝑓𝑦)))
215214biimprd 247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Lim 𝑦 → (𝑥 = (𝑓𝑦) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
216215adantld 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑦 → ((Lim 𝑦𝑥 = (𝑓𝑦)) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
217205, 210, 2163syld 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Lim 𝑦 → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
218217adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 ∈ V ∧ Lim 𝑦) → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
219214biimpd 228 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → 𝑥 = (𝑓𝑦)))
220219anc2li 554 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → (Lim 𝑦𝑥 = (𝑓𝑦))))
221 orc 865 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((Lim 𝑦𝑥 = (𝑓𝑦)) → ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
222221olcd 872 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((Lim 𝑦𝑥 = (𝑓𝑦)) → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
223220, 222syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))))
224223adantl 480 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 ∈ V ∧ Lim 𝑦) → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))))
225218, 224impbid 211 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ∈ V ∧ Lim 𝑦) → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
226164, 202, 2253jaoi 1425 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ (𝑦 ∈ V ∧ Lim 𝑦)) → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
227137, 226sylbi 216 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ On → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
228136, 227bitrid 282 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ On → (⟨𝑓, 𝑦⟩(((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
22953, 228syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (⟨𝑓, 𝑦⟩(((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
23026, 69brcnv 5880 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥Apply⟨𝑓, 𝑦⟩ ↔ ⟨𝑓, 𝑦⟩Apply𝑥)
23112, 39, 26brapply 35773 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑓, 𝑦⟩Apply𝑥𝑥 = (𝑓𝑦))
232230, 231bitri 274 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥Apply⟨𝑓, 𝑦⟩ ↔ 𝑥 = (𝑓𝑦))
233232a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (𝑥Apply⟨𝑓, 𝑦⟩ ↔ 𝑥 = (𝑓𝑦)))
234229, 233anbi12d 630 . . . . . . . . . . . . . . . . . . . . 21 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → ((⟨𝑓, 𝑦⟩(((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥𝑥Apply⟨𝑓, 𝑦⟩) ↔ (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ∧ 𝑥 = (𝑓𝑦))))
235234biancomd 462 . . . . . . . . . . . . . . . . . . . 20 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → ((⟨𝑓, 𝑦⟩(((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥𝑥Apply⟨𝑓, 𝑦⟩) ↔ (𝑥 = (𝑓𝑦) ∧ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
236235exbidv 1917 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (∃𝑥(⟨𝑓, 𝑦⟩(((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥𝑥Apply⟨𝑓, 𝑦⟩) ↔ ∃𝑥(𝑥 = (𝑓𝑦) ∧ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
237 df-br 5145 . . . . . . . . . . . . . . . . . . . 20 (𝑓 Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))
23869elfix 35738 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑓, 𝑦⟩ ∈ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))) ↔ ⟨𝑓, 𝑦⟩(Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))⟨𝑓, 𝑦⟩)
23969, 69brco 5868 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑓, 𝑦⟩(Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))⟨𝑓, 𝑦⟩ ↔ ∃𝑥(⟨𝑓, 𝑦⟩(((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥𝑥Apply⟨𝑓, 𝑦⟩))
240237, 238, 2393bitri 296 . . . . . . . . . . . . . . . . . . 19 (𝑓 Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦 ↔ ∃𝑥(⟨𝑓, 𝑦⟩(((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥𝑥Apply⟨𝑓, 𝑦⟩))
241 fvex 6904 . . . . . . . . . . . . . . . . . . . 20 (𝑓𝑦) ∈ V
242241eqvinc 3634 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ↔ ∃𝑥(𝑥 = (𝑓𝑦) ∧ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
243236, 240, 2423bitr4g 313 . . . . . . . . . . . . . . . . . 18 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (𝑓 Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦 ↔ (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
244243notbid 317 . . . . . . . . . . . . . . . . 17 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (¬ 𝑓 Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦 ↔ ¬ (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
2452443expia 1118 . . . . . . . . . . . . . . . 16 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (𝑦 ∈ dom 𝑓 → (¬ 𝑓 Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦 ↔ ¬ (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
246245pm5.32d 575 . . . . . . . . . . . . . . 15 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → ((𝑦 ∈ dom 𝑓 ∧ ¬ 𝑓 Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦) ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
247 annim 402 . . . . . . . . . . . . . . 15 ((𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))) ↔ ¬ (𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
248246, 247bitrdi 286 . . . . . . . . . . . . . 14 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → ((𝑦 ∈ dom 𝑓 ∧ ¬ 𝑓 Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦) ↔ ¬ (𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
24951, 248bitrid 282 . . . . . . . . . . . . 13 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (𝑓(( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦 ↔ ¬ (𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
250249exbidv 1917 . . . . . . . . . . . 12 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (∃𝑦 𝑓(( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦 ↔ ∃𝑦 ¬ (𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
251 exnal 1822 . . . . . . . . . . . 12 (∃𝑦 ¬ (𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))) ↔ ¬ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
252250, 251bitr2di 287 . . . . . . . . . . 11 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (¬ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))) ↔ ∃𝑦 𝑓(( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦))
25312eldm 5898 . . . . . . . . . . 11 (𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))) ↔ ∃𝑦 𝑓(( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦)
254252, 253bitr4di 288 . . . . . . . . . 10 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (¬ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))) ↔ 𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))))
255254con1bid 354 . . . . . . . . 9 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (¬ 𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))) ↔ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
256 df-ral 3052 . . . . . . . . 9 (∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ↔ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
257255, 256bitr4di 288 . . . . . . . 8 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (¬ 𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))) ↔ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
258257pm5.32i 573 . . . . . . 7 (((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ¬ 𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
259 anass 467 . . . . . . 7 (((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
260258, 259bitri 274 . . . . . 6 (((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ¬ 𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
26121, 37, 2603bitri 296 . . . . 5 (𝑓 ∈ (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
26219, 20, 2613bitr4ri 303 . . . 4 (𝑓 ∈ (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
263262eqabi 2862 . . 3 (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))}
264263unieqi 4918 . 2 (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))}
2651, 264eqtr4i 2757 1 rec(𝐹, 𝐴) = (( Funs ∩ (Domain “ On)) ∖ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  w3o 1083  w3a 1084  wal 1532   = wceq 1534  wex 1774  wcel 2099  {cab 2703  wne 2930  wral 3051  wrex 3060  Vcvv 3463  cdif 3944  cun 3945  cin 3946  c0 4323  ifcif 4524  {csn 4624  cop 4630   cuni 4906   class class class wbr 5144   I cid 5570   E cep 5576   × cxp 5671  ccnv 5672  dom cdm 5673  ran crn 5674  cres 5675  cima 5676  ccom 5677  Oncon0 6366  Lim wlim 6367  suc csuc 6368  Fun wfun 6538   Fn wfn 6539  cfv 6544  reccrdg 8429  pprodcpprod 35666   Bigcup cbigcup 35669   Fix cfix 35670   Limits climits 35671   Funs cfuns 35672  Imgcimg 35677  Domaincdomain 35678  Applycapply 35680  Succcsuccf 35683  FullFuncfullfn 35685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7736
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3366  df-rab 3421  df-v 3465  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-symdif 4242  df-nul 4324  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4907  df-iun 4996  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6303  df-ord 6369  df-on 6370  df-lim 6371  df-suc 6372  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7417  df-om 7867  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-txp 35689  df-pprod 35690  df-bigcup 35693  df-fix 35694  df-limits 35695  df-funs 35696  df-singleton 35697  df-singles 35698  df-image 35699  df-cart 35700  df-img 35701  df-domain 35702  df-cup 35704  df-succf 35707  df-apply 35708  df-funpart 35709  df-fullfun 35710
This theorem is referenced by: (None)
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