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Theorem dfrdg2 32021
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 7744 . . 3 (𝑖 = 𝐼 → rec(𝐹, 𝑖) = rec(𝐹, 𝐼))
2 ifeq1 4283 . . . . . . . . 9 (𝑖 = 𝐼 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
32eqeq2d 2816 . . . . . . . 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 3174 . . . . . . 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 616 . . . . . 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 3240 . . . . 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 2925 . . . 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 4640 . . 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 2821 . 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 7742 . . 3 rec(𝐹, 𝑖) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
11 dfrecs3 7705 . . 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 3394 . . . . . . . . . . . . 13 𝑓 ∈ V
1312resex 5648 . . . . . . . . . . . 12 (𝑓𝑦) ∈ V
14 eqeq1 2810 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓𝑦) → (𝑔 = ∅ ↔ (𝑓𝑦) = ∅))
15 relres 5629 . . . . . . . . . . . . . . . 16 Rel (𝑓𝑦)
16 reldm0 5544 . . . . . . . . . . . . . . . 16 (Rel (𝑓𝑦) → ((𝑓𝑦) = ∅ ↔ dom (𝑓𝑦) = ∅))
1715, 16ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑓𝑦) = ∅ ↔ dom (𝑓𝑦) = ∅)
1814, 17syl6bb 278 . . . . . . . . . . . . . 14 (𝑔 = (𝑓𝑦) → (𝑔 = ∅ ↔ dom (𝑓𝑦) = ∅))
19 dmeq 5525 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑓𝑦) → dom 𝑔 = dom (𝑓𝑦))
20 limeq 5948 . . . . . . . . . . . . . . . 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 5552 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑓𝑦) → ran 𝑔 = ran (𝑓𝑦))
23 df-ima 5324 . . . . . . . . . . . . . . . . 17 (𝑓𝑦) = ran (𝑓𝑦)
2422, 23syl6eqr 2858 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑓𝑦) → ran 𝑔 = (𝑓𝑦))
2524unieqd 4640 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓𝑦) → ran 𝑔 = (𝑓𝑦))
26 id 22 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑓𝑦) → 𝑔 = (𝑓𝑦))
2719unieqd 4640 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑓𝑦) → dom 𝑔 = dom (𝑓𝑦))
2826, 27fveq12d 6415 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑓𝑦) → (𝑔 dom 𝑔) = ((𝑓𝑦)‘ dom (𝑓𝑦)))
2928fveq2d 6412 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓𝑦) → (𝐹‘(𝑔 dom 𝑔)) = (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))))
3021, 25, 29ifbieq12d 4306 . . . . . . . . . . . . . 14 (𝑔 = (𝑓𝑦) → if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦)))))
3118, 30ifbieq2d 4304 . . . . . . . . . . . . 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 2806 . . . . . . . . . . . . 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 3394 . . . . . . . . . . . . . 14 𝑖 ∈ V
34 imaexg 7333 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ V → (𝑓𝑦) ∈ V)
3512, 34ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑓𝑦) ∈ V
3635uniex 7183 . . . . . . . . . . . . . . 15 (𝑓𝑦) ∈ V
37 fvex 6421 . . . . . . . . . . . . . . 15 (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))) ∈ V
3836, 37ifex 4327 . . . . . . . . . . . . . 14 if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦)))) ∈ V
3933, 38ifex 4327 . . . . . . . . . . . . 13 if(dom (𝑓𝑦) = ∅, 𝑖, if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))))) ∈ V
4031, 32, 39fvmpt 6503 . . . . . . . . . . . 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 5622 . . . . . . . . . . . . 13 dom (𝑓𝑦) = (𝑦 ∩ dom 𝑓)
43 onelss 5979 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
4443imp 395 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
45443adant2 1154 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → 𝑦𝑥)
46 fndm 6201 . . . . . . . . . . . . . . . 16 (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
47463ad2ant2 1157 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → dom 𝑓 = 𝑥)
4845, 47sseqtr4d 3839 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → 𝑦 ⊆ dom 𝑓)
49 df-ss 3783 . . . . . . . . . . . . . 14 (𝑦 ⊆ dom 𝑓 ↔ (𝑦 ∩ dom 𝑓) = 𝑦)
5048, 49sylib 209 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → (𝑦 ∩ dom 𝑓) = 𝑦)
5142, 50syl5eq 2852 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → dom (𝑓𝑦) = 𝑦)
52 eqeq1 2810 . . . . . . . . . . . . . 14 (dom (𝑓𝑦) = 𝑦 → (dom (𝑓𝑦) = ∅ ↔ 𝑦 = ∅))
53 limeq 5948 . . . . . . . . . . . . . . 15 (dom (𝑓𝑦) = 𝑦 → (Lim dom (𝑓𝑦) ↔ Lim 𝑦))
54 unieq 4638 . . . . . . . . . . . . . . . . 17 (dom (𝑓𝑦) = 𝑦 dom (𝑓𝑦) = 𝑦)
5554fveq2d 6412 . . . . . . . . . . . . . . . 16 (dom (𝑓𝑦) = 𝑦 → ((𝑓𝑦)‘ dom (𝑓𝑦)) = ((𝑓𝑦)‘ 𝑦))
5655fveq2d 6412 . . . . . . . . . . . . . . 15 (dom (𝑓𝑦) = 𝑦 → (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))) = (𝐹‘((𝑓𝑦)‘ 𝑦)))
5753, 56ifbieq2d 4304 . . . . . . . . . . . . . 14 (dom (𝑓𝑦) = 𝑦 → if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))))
5852, 57ifbieq2d 4304 . . . . . . . . . . . . 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 5961 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
60 eloni 5946 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → Ord 𝑦)
6159, 60syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦𝑥) → Ord 𝑦)
62613adant2 1154 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → Ord 𝑦)
63 ordzsl 7275 . . . . . . . . . . . . . . 15 (Ord 𝑦 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ Lim 𝑦))
64 iftrue 4285 . . . . . . . . . . . . . . . . 17 (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = 𝑖)
65 iftrue 4285 . . . . . . . . . . . . . . . . 17 (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = 𝑖)
6664, 65eqtr4d 2843 . . . . . . . . . . . . . . . 16 (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
67 vex 3394 . . . . . . . . . . . . . . . . . . . . . . 23 𝑧 ∈ V
6867sucid 6017 . . . . . . . . . . . . . . . . . . . . . 22 𝑧 ∈ suc 𝑧
69 fvres 6427 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ suc 𝑧 → ((𝑓 ↾ suc 𝑧)‘𝑧) = (𝑓𝑧))
7068, 69ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ↾ suc 𝑧)‘𝑧) = (𝑓𝑧)
71 eloni 5946 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ On → Ord 𝑧)
72 ordunisuc 7262 . . . . . . . . . . . . . . . . . . . . . . 23 (Ord 𝑧 suc 𝑧 = 𝑧)
7371, 72syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ On → suc 𝑧 = 𝑧)
7473fveq2d 6412 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘ suc 𝑧) = ((𝑓 ↾ suc 𝑧)‘𝑧))
7573fveq2d 6412 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ On → (𝑓 suc 𝑧) = (𝑓𝑧))
7670, 74, 753eqtr4a 2866 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘ suc 𝑧) = (𝑓 suc 𝑧))
7776fveq2d 6412 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ On → (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)) = (𝐹‘(𝑓 suc 𝑧)))
78 nsuceq0 6018 . . . . . . . . . . . . . . . . . . . . . 22 suc 𝑧 ≠ ∅
7978neii 2980 . . . . . . . . . . . . . . . . . . . . 21 ¬ suc 𝑧 = ∅
8079iffalsei 4289 . . . . . . . . . . . . . . . . . . . 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 7272 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ V → ¬ Lim suc 𝑧)
82 iffalse 4288 . . . . . . . . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . . . . . . 19 if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)))) = (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧))
8579iffalsei 4289 . . . . . . . . . . . . . . . . . . . 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 4288 . . . . . . . . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . . . . . . 19 if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))) = (𝐹‘(𝑓 suc 𝑧))
8977, 84, 883eqtr4g 2865 . . . . . . . . . . . . . . . . . 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 2810 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
91 limeq 5948 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = suc 𝑧 → (Lim 𝑦 ↔ Lim suc 𝑧))
92 reseq2 5592 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = suc 𝑧 → (𝑓𝑦) = (𝑓 ↾ suc 𝑧))
93 unieq 4638 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = suc 𝑧 𝑦 = suc 𝑧)
9492, 93fveq12d 6415 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = suc 𝑧 → ((𝑓𝑦)‘ 𝑦) = ((𝑓 ↾ suc 𝑧)‘ suc 𝑧))
9594fveq2d 6412 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = suc 𝑧 → (𝐹‘((𝑓𝑦)‘ 𝑦)) = (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)))
9691, 95ifbieq2d 4304 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = suc 𝑧 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))) = if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧))))
9790, 96ifbieq2d 4304 . . . . . . . . . . . . . . . . . . 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 6412 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = suc 𝑧 → (𝑓 𝑦) = (𝑓 suc 𝑧))
9998fveq2d 6412 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = suc 𝑧 → (𝐹‘(𝑓 𝑦)) = (𝐹‘(𝑓 suc 𝑧)))
10091, 99ifbieq2d 4304 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = suc 𝑧 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))) = if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧))))
10190, 100ifbieq2d 4304 . . . . . . . . . . . . . . . . . . 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 2821 . . . . . . . . . . . . . . . . . 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 238 . . . . . . . . . . . . . . . . 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 3215 . . . . . . . . . . . . . . . 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 4285 . . . . . . . . . . . . . . . . . 18 (Lim 𝑦 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))) = (𝑓𝑦))
106 df-lim 5941 . . . . . . . . . . . . . . . . . . . . 21 (Lim 𝑦 ↔ (Ord 𝑦𝑦 ≠ ∅ ∧ 𝑦 = 𝑦))
107106simp2bi 1169 . . . . . . . . . . . . . . . . . . . 20 (Lim 𝑦𝑦 ≠ ∅)
108107neneqd 2983 . . . . . . . . . . . . . . . . . . 19 (Lim 𝑦 → ¬ 𝑦 = ∅)
109108iffalsed 4290 . . . . . . . . . . . . . . . . . 18 (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))))
110 iftrue 4285 . . . . . . . . . . . . . . . . . 18 (Lim 𝑦 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))) = (𝑓𝑦))
111105, 109, 1103eqtr4d 2850 . . . . . . . . . . . . . . . . 17 (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))
112108iffalsed 4290 . . . . . . . . . . . . . . . . 17 (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))
113111, 112eqtr4d 2843 . . . . . . . . . . . . . . . 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 1545 . . . . . . . . . . . . . . 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 208 . . . . . . . . . . . . . 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 2862 . . . . . . . . . . . 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 670 . . . . . . . . . . 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 2852 . . . . . . . . . 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 2816 . . . . . . . . 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 1140 . . . . . . . 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 3173 . . . . . . 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 570 . . . . . 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 3228 . . . . 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 2923 . . . 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 4639 . . 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 2832 . 2 rec(𝐹, 𝑖) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))}
1289, 127vtoclg 3459 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 197  wa 384  w3o 1099  w3a 1100   = wceq 1637  wcel 2156  {cab 2792  wne 2978  wral 3096  wrex 3097  Vcvv 3391  cin 3768  wss 3769  c0 4116  ifcif 4279   cuni 4630  cmpt 4923  dom cdm 5311  ran crn 5312  cres 5313  cima 5314  Rel wrel 5316  Ord word 5935  Oncon0 5936  Lim wlim 5937  suc csuc 5938   Fn wfn 6096  cfv 6101  recscrecs 7703  reccrdg 7741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-pss 3785  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-tp 4375  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-tr 4947  df-id 5219  df-eprel 5224  df-po 5232  df-so 5233  df-fr 5270  df-we 5272  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-pred 5893  df-ord 5939  df-on 5940  df-lim 5941  df-suc 5942  df-iota 6064  df-fun 6103  df-fn 6104  df-fv 6109  df-wrecs 7642  df-recs 7704  df-rdg 7742
This theorem is referenced by:  dfrdg3  32022
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