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Theorem dfrdg2 31423
Description: Alternate definition of the recursive function generator when 𝐼 is a set. (Contributed by Scott Fenton, 26-Mar-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
dfrdg2 (𝐼𝑉 → rec(𝐹, 𝐼) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))})
Distinct variable groups:   𝑓,𝐹,𝑥,𝑦   𝑓,𝐼,𝑥,𝑦
Allowed substitution hints:   𝑉(𝑥,𝑦,𝑓)

Proof of Theorem dfrdg2
Dummy variables 𝑔 𝑖 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rdgeq2 7456 . . 3 (𝑖 = 𝐼 → rec(𝐹, 𝑖) = rec(𝐹, 𝐼))
2 ifeq1 4064 . . . . . . . . 9 (𝑖 = 𝐼 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
32eqeq2d 2631 . . . . . . . 8 (𝑖 = 𝐼 → ((𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ↔ (𝑓𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
43ralbidv 2980 . . . . . . 7 (𝑖 = 𝐼 → (∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ↔ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
54anbi2d 739 . . . . . 6 (𝑖 = 𝐼 → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))) ↔ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
65rexbidv 3045 . . . . 5 (𝑖 = 𝐼 → (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))) ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
76abbidv 2738 . . . 4 (𝑖 = 𝐼 → {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))})
87unieqd 4414 . . 3 (𝑖 = 𝐼 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))})
91, 8eqeq12d 2636 . 2 (𝑖 = 𝐼 → (rec(𝐹, 𝑖) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))} ↔ rec(𝐹, 𝐼) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))}))
10 df-rdg 7454 . . 3 rec(𝐹, 𝑖) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
11 dfrecs3 7417 . . 3 recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(𝑓𝑦)))}
12 vex 3189 . . . . . . . . . . . . 13 𝑓 ∈ V
1312resex 5404 . . . . . . . . . . . 12 (𝑓𝑦) ∈ V
14 eqeq1 2625 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓𝑦) → (𝑔 = ∅ ↔ (𝑓𝑦) = ∅))
15 relres 5387 . . . . . . . . . . . . . . . 16 Rel (𝑓𝑦)
16 reldm0 5305 . . . . . . . . . . . . . . . 16 (Rel (𝑓𝑦) → ((𝑓𝑦) = ∅ ↔ dom (𝑓𝑦) = ∅))
1715, 16ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑓𝑦) = ∅ ↔ dom (𝑓𝑦) = ∅)
1814, 17syl6bb 276 . . . . . . . . . . . . . 14 (𝑔 = (𝑓𝑦) → (𝑔 = ∅ ↔ dom (𝑓𝑦) = ∅))
19 dmeq 5286 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑓𝑦) → dom 𝑔 = dom (𝑓𝑦))
20 limeq 5696 . . . . . . . . . . . . . . . 16 (dom 𝑔 = dom (𝑓𝑦) → (Lim dom 𝑔 ↔ Lim dom (𝑓𝑦)))
2119, 20syl 17 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓𝑦) → (Lim dom 𝑔 ↔ Lim dom (𝑓𝑦)))
22 rneq 5313 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑓𝑦) → ran 𝑔 = ran (𝑓𝑦))
23 df-ima 5089 . . . . . . . . . . . . . . . . 17 (𝑓𝑦) = ran (𝑓𝑦)
2422, 23syl6eqr 2673 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑓𝑦) → ran 𝑔 = (𝑓𝑦))
2524unieqd 4414 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓𝑦) → ran 𝑔 = (𝑓𝑦))
26 id 22 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑓𝑦) → 𝑔 = (𝑓𝑦))
2719unieqd 4414 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑓𝑦) → dom 𝑔 = dom (𝑓𝑦))
2826, 27fveq12d 6156 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑓𝑦) → (𝑔 dom 𝑔) = ((𝑓𝑦)‘ dom (𝑓𝑦)))
2928fveq2d 6154 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓𝑦) → (𝐹‘(𝑔 dom 𝑔)) = (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))))
3021, 25, 29ifbieq12d 4087 . . . . . . . . . . . . . 14 (𝑔 = (𝑓𝑦) → if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦)))))
3118, 30ifbieq2d 4085 . . . . . . . . . . . . 13 (𝑔 = (𝑓𝑦) → if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))) = if(dom (𝑓𝑦) = ∅, 𝑖, if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))))))
32 eqid 2621 . . . . . . . . . . . . 13 (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))) = (𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))
33 vex 3189 . . . . . . . . . . . . . 14 𝑖 ∈ V
34 imaexg 7053 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ V → (𝑓𝑦) ∈ V)
3512, 34ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑓𝑦) ∈ V
3635uniex 6909 . . . . . . . . . . . . . . 15 (𝑓𝑦) ∈ V
37 fvex 6160 . . . . . . . . . . . . . . 15 (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))) ∈ V
3836, 37ifex 4130 . . . . . . . . . . . . . 14 if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦)))) ∈ V
3933, 38ifex 4130 . . . . . . . . . . . . 13 if(dom (𝑓𝑦) = ∅, 𝑖, if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))))) ∈ V
4031, 32, 39fvmpt 6241 . . . . . . . . . . . 12 ((𝑓𝑦) ∈ V → ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(𝑓𝑦)) = if(dom (𝑓𝑦) = ∅, 𝑖, if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))))))
4113, 40ax-mp 5 . . . . . . . . . . 11 ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(𝑓𝑦)) = if(dom (𝑓𝑦) = ∅, 𝑖, if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦)))))
42 dmres 5380 . . . . . . . . . . . . 13 dom (𝑓𝑦) = (𝑦 ∩ dom 𝑓)
43 onelss 5727 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
4443imp 445 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
45443adant2 1078 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → 𝑦𝑥)
46 fndm 5950 . . . . . . . . . . . . . . . 16 (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
47463ad2ant2 1081 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → dom 𝑓 = 𝑥)
4845, 47sseqtr4d 3623 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → 𝑦 ⊆ dom 𝑓)
49 df-ss 3570 . . . . . . . . . . . . . 14 (𝑦 ⊆ dom 𝑓 ↔ (𝑦 ∩ dom 𝑓) = 𝑦)
5048, 49sylib 208 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → (𝑦 ∩ dom 𝑓) = 𝑦)
5142, 50syl5eq 2667 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → dom (𝑓𝑦) = 𝑦)
52 eqeq1 2625 . . . . . . . . . . . . . 14 (dom (𝑓𝑦) = 𝑦 → (dom (𝑓𝑦) = ∅ ↔ 𝑦 = ∅))
53 limeq 5696 . . . . . . . . . . . . . . 15 (dom (𝑓𝑦) = 𝑦 → (Lim dom (𝑓𝑦) ↔ Lim 𝑦))
54 unieq 4412 . . . . . . . . . . . . . . . . 17 (dom (𝑓𝑦) = 𝑦 dom (𝑓𝑦) = 𝑦)
5554fveq2d 6154 . . . . . . . . . . . . . . . 16 (dom (𝑓𝑦) = 𝑦 → ((𝑓𝑦)‘ dom (𝑓𝑦)) = ((𝑓𝑦)‘ 𝑦))
5655fveq2d 6154 . . . . . . . . . . . . . . 15 (dom (𝑓𝑦) = 𝑦 → (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))) = (𝐹‘((𝑓𝑦)‘ 𝑦)))
5753, 56ifbieq2d 4085 . . . . . . . . . . . . . 14 (dom (𝑓𝑦) = 𝑦 → if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))))
5852, 57ifbieq2d 4085 . . . . . . . . . . . . 13 (dom (𝑓𝑦) = 𝑦 → if(dom (𝑓𝑦) = ∅, 𝑖, if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))))
59 onelon 5709 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
60 eloni 5694 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → Ord 𝑦)
6159, 60syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦𝑥) → Ord 𝑦)
62613adant2 1078 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → Ord 𝑦)
63 ordzsl 6995 . . . . . . . . . . . . . . 15 (Ord 𝑦 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ Lim 𝑦))
64 iftrue 4066 . . . . . . . . . . . . . . . . 17 (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = 𝑖)
65 iftrue 4066 . . . . . . . . . . . . . . . . 17 (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = 𝑖)
6664, 65eqtr4d 2658 . . . . . . . . . . . . . . . 16 (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
67 vex 3189 . . . . . . . . . . . . . . . . . . . . . . 23 𝑧 ∈ V
6867sucid 5765 . . . . . . . . . . . . . . . . . . . . . 22 𝑧 ∈ suc 𝑧
69 fvres 6166 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ suc 𝑧 → ((𝑓 ↾ suc 𝑧)‘𝑧) = (𝑓𝑧))
7068, 69ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ↾ suc 𝑧)‘𝑧) = (𝑓𝑧)
71 eloni 5694 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ On → Ord 𝑧)
72 ordunisuc 6982 . . . . . . . . . . . . . . . . . . . . . . 23 (Ord 𝑧 suc 𝑧 = 𝑧)
7371, 72syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ On → suc 𝑧 = 𝑧)
7473fveq2d 6154 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘ suc 𝑧) = ((𝑓 ↾ suc 𝑧)‘𝑧))
7573fveq2d 6154 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ On → (𝑓 suc 𝑧) = (𝑓𝑧))
7670, 74, 753eqtr4a 2681 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘ suc 𝑧) = (𝑓 suc 𝑧))
7776fveq2d 6154 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ On → (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)) = (𝐹‘(𝑓 suc 𝑧)))
78 nsuceq0 5766 . . . . . . . . . . . . . . . . . . . . . 22 suc 𝑧 ≠ ∅
7978neii 2792 . . . . . . . . . . . . . . . . . . . . 21 ¬ suc 𝑧 = ∅
8079iffalsei 4070 . . . . . . . . . . . . . . . . . . . 20 if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)))) = if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)))
81 nlimsucg 6992 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ V → ¬ Lim suc 𝑧)
82 iffalse 4069 . . . . . . . . . . . . . . . . . . . . 21 (¬ Lim suc 𝑧 → if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧))) = (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)))
8367, 81, 82mp2b 10 . . . . . . . . . . . . . . . . . . . 20 if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧))) = (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧))
8480, 83eqtri 2643 . . . . . . . . . . . . . . . . . . 19 if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)))) = (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧))
8579iffalsei 4070 . . . . . . . . . . . . . . . . . . . 20 if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))) = if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))
86 iffalse 4069 . . . . . . . . . . . . . . . . . . . . 21 (¬ Lim suc 𝑧 → if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧))) = (𝐹‘(𝑓 suc 𝑧)))
8767, 81, 86mp2b 10 . . . . . . . . . . . . . . . . . . . 20 if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧))) = (𝐹‘(𝑓 suc 𝑧))
8885, 87eqtri 2643 . . . . . . . . . . . . . . . . . . 19 if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))) = (𝐹‘(𝑓 suc 𝑧))
8977, 84, 883eqtr4g 2680 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ On → if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)))) = if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))))
90 eqeq1 2625 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
91 limeq 5696 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = suc 𝑧 → (Lim 𝑦 ↔ Lim suc 𝑧))
92 reseq2 5353 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = suc 𝑧 → (𝑓𝑦) = (𝑓 ↾ suc 𝑧))
93 unieq 4412 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = suc 𝑧 𝑦 = suc 𝑧)
9492, 93fveq12d 6156 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = suc 𝑧 → ((𝑓𝑦)‘ 𝑦) = ((𝑓 ↾ suc 𝑧)‘ suc 𝑧))
9594fveq2d 6154 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = suc 𝑧 → (𝐹‘((𝑓𝑦)‘ 𝑦)) = (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)))
9691, 95ifbieq2d 4085 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = suc 𝑧 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))) = if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧))))
9790, 96ifbieq2d 4085 . . . . . . . . . . . . . . . . . . 19 (𝑦 = suc 𝑧 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)))))
9893fveq2d 6154 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = suc 𝑧 → (𝑓 𝑦) = (𝑓 suc 𝑧))
9998fveq2d 6154 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = suc 𝑧 → (𝐹‘(𝑓 𝑦)) = (𝐹‘(𝑓 suc 𝑧)))
10091, 99ifbieq2d 4085 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = suc 𝑧 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))) = if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧))))
10190, 100ifbieq2d 4085 . . . . . . . . . . . . . . . . . . 19 (𝑦 = suc 𝑧 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))))
10297, 101eqeq12d 2636 . . . . . . . . . . . . . . . . . 18 (𝑦 = suc 𝑧 → (if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ↔ if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)))) = if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧))))))
10389, 102syl5ibrcom 237 . . . . . . . . . . . . . . . . 17 (𝑧 ∈ On → (𝑦 = suc 𝑧 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
104103rexlimiv 3020 . . . . . . . . . . . . . . . 16 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
105 iftrue 4066 . . . . . . . . . . . . . . . . . 18 (Lim 𝑦 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))) = (𝑓𝑦))
106 df-lim 5689 . . . . . . . . . . . . . . . . . . . . 21 (Lim 𝑦 ↔ (Ord 𝑦𝑦 ≠ ∅ ∧ 𝑦 = 𝑦))
107106simp2bi 1075 . . . . . . . . . . . . . . . . . . . 20 (Lim 𝑦𝑦 ≠ ∅)
108107neneqd 2795 . . . . . . . . . . . . . . . . . . 19 (Lim 𝑦 → ¬ 𝑦 = ∅)
109108iffalsed 4071 . . . . . . . . . . . . . . . . . 18 (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))))
110 iftrue 4066 . . . . . . . . . . . . . . . . . 18 (Lim 𝑦 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))) = (𝑓𝑦))
111105, 109, 1103eqtr4d 2665 . . . . . . . . . . . . . . . . 17 (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))
112108iffalsed 4071 . . . . . . . . . . . . . . . . 17 (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))
113111, 112eqtr4d 2658 . . . . . . . . . . . . . . . 16 (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
11466, 104, 1133jaoi 1388 . . . . . . . . . . . . . . 15 ((𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ Lim 𝑦) → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
11563, 114sylbi 207 . . . . . . . . . . . . . 14 (Ord 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
11662, 115syl 17 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
11758, 116sylan9eqr 2677 . . . . . . . . . . . 12 (((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) ∧ dom (𝑓𝑦) = 𝑦) → if(dom (𝑓𝑦) = ∅, 𝑖, if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
11851, 117mpdan 701 . . . . . . . . . . 11 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → if(dom (𝑓𝑦) = ∅, 𝑖, if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
11941, 118syl5eq 2667 . . . . . . . . . 10 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(𝑓𝑦)) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
120119eqeq2d 2631 . . . . . . . . 9 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → ((𝑓𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(𝑓𝑦)) ↔ (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
1211203expa 1262 . . . . . . . 8 (((𝑥 ∈ On ∧ 𝑓 Fn 𝑥) ∧ 𝑦𝑥) → ((𝑓𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(𝑓𝑦)) ↔ (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
122121ralbidva 2979 . . . . . . 7 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥) → (∀𝑦𝑥 (𝑓𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(𝑓𝑦)) ↔ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
123122pm5.32da 672 . . . . . 6 (𝑥 ∈ On → ((𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(𝑓𝑦))) ↔ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
124123rexbiia 3033 . . . . 5 (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(𝑓𝑦))) ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
125124abbii 2736 . . . 4 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))}
126125unieqi 4413 . . 3 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = ((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔)))))‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))}
12710, 11, 1263eqtri 2647 . 2 rec(𝐹, 𝑖) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))}
1289, 127vtoclg 3252 1 (𝐼𝑉 → rec(𝐹, 𝐼) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3o 1035  w3a 1036   = wceq 1480  wcel 1987  {cab 2607  wne 2790  wral 2907  wrex 2908  Vcvv 3186  cin 3555  wss 3556  c0 3893  ifcif 4060   cuni 4404  cmpt 4675  dom cdm 5076  ran crn 5077  cres 5078  cima 5079  Rel wrel 5081  Ord word 5683  Oncon0 5684  Lim wlim 5685  suc csuc 5686   Fn wfn 5844  cfv 5849  recscrecs 7415  reccrdg 7453
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869  ax-un 6905
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-pss 3572  df-nul 3894  df-if 4061  df-pw 4134  df-sn 4151  df-pr 4153  df-tp 4155  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-tr 4715  df-eprel 4987  df-id 4991  df-po 4997  df-so 4998  df-fr 5035  df-we 5037  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-pred 5641  df-ord 5687  df-on 5688  df-lim 5689  df-suc 5690  df-iota 5812  df-fun 5851  df-fn 5852  df-fv 5857  df-wrecs 7355  df-recs 7416  df-rdg 7454
This theorem is referenced by:  dfrdg3  31424
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