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Theorem dfrdg2 34370
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 8358 . . 3 (𝑖 = 𝐼 → rec(𝐹, 𝑖) = rec(𝐹, 𝐼))
2 ifeq1 4490 . . . . . . . . 9 (𝑖 = 𝐼 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
32eqeq2d 2747 . . . . . . . 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 629 . . . . . 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 3175 . . . . 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 2805 . . . 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 4879 . . 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 2752 . 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 8356 . . 3 rec(𝐹, 𝑖) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
11 dfrecs3 8318 . . 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 3449 . . . . . . . . . . . . 13 𝑓 ∈ V
1312resex 5985 . . . . . . . . . . . 12 (𝑓𝑦) ∈ V
14 eqeq1 2740 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓𝑦) → (𝑔 = ∅ ↔ (𝑓𝑦) = ∅))
15 relres 5966 . . . . . . . . . . . . . . . 16 Rel (𝑓𝑦)
16 reldm0 5883 . . . . . . . . . . . . . . . 16 (Rel (𝑓𝑦) → ((𝑓𝑦) = ∅ ↔ dom (𝑓𝑦) = ∅))
1715, 16ax-mp 5 . . . . . . . . . . . . . . 15 ((𝑓𝑦) = ∅ ↔ dom (𝑓𝑦) = ∅)
1814, 17bitrdi 286 . . . . . . . . . . . . . 14 (𝑔 = (𝑓𝑦) → (𝑔 = ∅ ↔ dom (𝑓𝑦) = ∅))
19 dmeq 5859 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑓𝑦) → dom 𝑔 = dom (𝑓𝑦))
20 limeq 6329 . . . . . . . . . . . . . . . 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 5891 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑓𝑦) → ran 𝑔 = ran (𝑓𝑦))
23 df-ima 5646 . . . . . . . . . . . . . . . . 17 (𝑓𝑦) = ran (𝑓𝑦)
2422, 23eqtr4di 2794 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑓𝑦) → ran 𝑔 = (𝑓𝑦))
2524unieqd 4879 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓𝑦) → ran 𝑔 = (𝑓𝑦))
26 id 22 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑓𝑦) → 𝑔 = (𝑓𝑦))
2719unieqd 4879 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑓𝑦) → dom 𝑔 = dom (𝑓𝑦))
2826, 27fveq12d 6849 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑓𝑦) → (𝑔 dom 𝑔) = ((𝑓𝑦)‘ dom (𝑓𝑦)))
2928fveq2d 6846 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓𝑦) → (𝐹‘(𝑔 dom 𝑔)) = (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))))
3021, 25, 29ifbieq12d 4514 . . . . . . . . . . . . . 14 (𝑔 = (𝑓𝑦) → if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦)))))
3118, 30ifbieq2d 4512 . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . 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 3449 . . . . . . . . . . . . . 14 𝑖 ∈ V
34 imaexg 7852 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ V → (𝑓𝑦) ∈ V)
3512, 34ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑓𝑦) ∈ V
3635uniex 7678 . . . . . . . . . . . . . . 15 (𝑓𝑦) ∈ V
37 fvex 6855 . . . . . . . . . . . . . . 15 (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))) ∈ V
3836, 37ifex 4536 . . . . . . . . . . . . . 14 if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦)))) ∈ V
3933, 38ifex 4536 . . . . . . . . . . . . 13 if(dom (𝑓𝑦) = ∅, 𝑖, if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))))) ∈ V
4031, 32, 39fvmpt 6948 . . . . . . . . . . . 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 5959 . . . . . . . . . . . . 13 dom (𝑓𝑦) = (𝑦 ∩ dom 𝑓)
43 onelss 6359 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
4443imp 407 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
45443adant2 1131 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → 𝑦𝑥)
46 fndm 6605 . . . . . . . . . . . . . . . 16 (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
47463ad2ant2 1134 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → dom 𝑓 = 𝑥)
4845, 47sseqtrrd 3985 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → 𝑦 ⊆ dom 𝑓)
49 df-ss 3927 . . . . . . . . . . . . . 14 (𝑦 ⊆ dom 𝑓 ↔ (𝑦 ∩ dom 𝑓) = 𝑦)
5048, 49sylib 217 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → (𝑦 ∩ dom 𝑓) = 𝑦)
5142, 50eqtrid 2788 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → dom (𝑓𝑦) = 𝑦)
52 eqeq1 2740 . . . . . . . . . . . . . 14 (dom (𝑓𝑦) = 𝑦 → (dom (𝑓𝑦) = ∅ ↔ 𝑦 = ∅))
53 limeq 6329 . . . . . . . . . . . . . . 15 (dom (𝑓𝑦) = 𝑦 → (Lim dom (𝑓𝑦) ↔ Lim 𝑦))
54 unieq 4876 . . . . . . . . . . . . . . . . 17 (dom (𝑓𝑦) = 𝑦 dom (𝑓𝑦) = 𝑦)
5554fveq2d 6846 . . . . . . . . . . . . . . . 16 (dom (𝑓𝑦) = 𝑦 → ((𝑓𝑦)‘ dom (𝑓𝑦)) = ((𝑓𝑦)‘ 𝑦))
5655fveq2d 6846 . . . . . . . . . . . . . . 15 (dom (𝑓𝑦) = 𝑦 → (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))) = (𝐹‘((𝑓𝑦)‘ 𝑦)))
5753, 56ifbieq2d 4512 . . . . . . . . . . . . . 14 (dom (𝑓𝑦) = 𝑦 → if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))))
5852, 57ifbieq2d 4512 . . . . . . . . . . . . 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 6342 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
60 eloni 6327 . . . . . . . . . . . . . . . 16 (𝑦 ∈ On → Ord 𝑦)
6159, 60syl 17 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦𝑥) → Ord 𝑦)
62613adant2 1131 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → Ord 𝑦)
63 ordzsl 7781 . . . . . . . . . . . . . . 15 (Ord 𝑦 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ Lim 𝑦))
64 iftrue 4492 . . . . . . . . . . . . . . . . 17 (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = 𝑖)
65 iftrue 4492 . . . . . . . . . . . . . . . . 17 (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = 𝑖)
6664, 65eqtr4d 2779 . . . . . . . . . . . . . . . 16 (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
67 vex 3449 . . . . . . . . . . . . . . . . . . . . . . 23 𝑧 ∈ V
6867sucid 6399 . . . . . . . . . . . . . . . . . . . . . 22 𝑧 ∈ suc 𝑧
69 fvres 6861 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ suc 𝑧 → ((𝑓 ↾ suc 𝑧)‘𝑧) = (𝑓𝑧))
7068, 69ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ↾ suc 𝑧)‘𝑧) = (𝑓𝑧)
71 eloni 6327 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ On → Ord 𝑧)
72 ordunisuc 7767 . . . . . . . . . . . . . . . . . . . . . . 23 (Ord 𝑧 suc 𝑧 = 𝑧)
7371, 72syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ On → suc 𝑧 = 𝑧)
7473fveq2d 6846 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘ suc 𝑧) = ((𝑓 ↾ suc 𝑧)‘𝑧))
7573fveq2d 6846 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ On → (𝑓 suc 𝑧) = (𝑓𝑧))
7670, 74, 753eqtr4a 2802 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘ suc 𝑧) = (𝑓 suc 𝑧))
7776fveq2d 6846 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ On → (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)) = (𝐹‘(𝑓 suc 𝑧)))
78 nsuceq0 6400 . . . . . . . . . . . . . . . . . . . . . 22 suc 𝑧 ≠ ∅
7978neii 2945 . . . . . . . . . . . . . . . . . . . . 21 ¬ suc 𝑧 = ∅
8079iffalsei 4496 . . . . . . . . . . . . . . . . . . . 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 7778 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ V → ¬ Lim suc 𝑧)
82 iffalse 4495 . . . . . . . . . . . . . . . . . . . . 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 2764 . . . . . . . . . . . . . . . . . . 19 if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)))) = (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧))
8579iffalsei 4496 . . . . . . . . . . . . . . . . . . . 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 4495 . . . . . . . . . . . . . . . . . . . . 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 2764 . . . . . . . . . . . . . . . . . . 19 if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))) = (𝐹‘(𝑓 suc 𝑧))
8977, 84, 883eqtr4g 2801 . . . . . . . . . . . . . . . . . 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 2740 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
91 limeq 6329 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = suc 𝑧 → (Lim 𝑦 ↔ Lim suc 𝑧))
92 reseq2 5932 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = suc 𝑧 → (𝑓𝑦) = (𝑓 ↾ suc 𝑧))
93 unieq 4876 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = suc 𝑧 𝑦 = suc 𝑧)
9492, 93fveq12d 6849 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = suc 𝑧 → ((𝑓𝑦)‘ 𝑦) = ((𝑓 ↾ suc 𝑧)‘ suc 𝑧))
9594fveq2d 6846 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = suc 𝑧 → (𝐹‘((𝑓𝑦)‘ 𝑦)) = (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)))
9691, 95ifbieq2d 4512 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = suc 𝑧 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))) = if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧))))
9790, 96ifbieq2d 4512 . . . . . . . . . . . . . . . . . . 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 6846 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = suc 𝑧 → (𝑓 𝑦) = (𝑓 suc 𝑧))
9998fveq2d 6846 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = suc 𝑧 → (𝐹‘(𝑓 𝑦)) = (𝐹‘(𝑓 suc 𝑧)))
10091, 99ifbieq2d 4512 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = suc 𝑧 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))) = if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧))))
10190, 100ifbieq2d 4512 . . . . . . . . . . . . . . . . . . 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 2752 . . . . . . . . . . . . . . . . . 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 246 . . . . . . . . . . . . . . . . 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 3145 . . . . . . . . . . . . . . . 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 4492 . . . . . . . . . . . . . . . . . 18 (Lim 𝑦 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))) = (𝑓𝑦))
106 df-lim 6322 . . . . . . . . . . . . . . . . . . . . 21 (Lim 𝑦 ↔ (Ord 𝑦𝑦 ≠ ∅ ∧ 𝑦 = 𝑦))
107106simp2bi 1146 . . . . . . . . . . . . . . . . . . . 20 (Lim 𝑦𝑦 ≠ ∅)
108107neneqd 2948 . . . . . . . . . . . . . . . . . . 19 (Lim 𝑦 → ¬ 𝑦 = ∅)
109108iffalsed 4497 . . . . . . . . . . . . . . . . . 18 (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))))
110 iftrue 4492 . . . . . . . . . . . . . . . . . 18 (Lim 𝑦 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))) = (𝑓𝑦))
111105, 109, 1103eqtr4d 2786 . . . . . . . . . . . . . . . . 17 (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))
112108iffalsed 4497 . . . . . . . . . . . . . . . . 17 (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))
113111, 112eqtr4d 2779 . . . . . . . . . . . . . . . 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 1427 . . . . . . . . . . . . . . 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 216 . . . . . . . . . . . . . 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 2798 . . . . . . . . . . . 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 685 . . . . . . . . . . 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, 118eqtrid 2788 . . . . . . . . . 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 2747 . . . . . . . . 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 1118 . . . . . . . 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 3172 . . . . . . 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 579 . . . . . 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 3095 . . . . 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 2806 . . . 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 4878 . . 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 2768 . 2 rec(𝐹, 𝑖) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))}
1289, 127vtoclg 3525 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 205  wa 396  w3o 1086  w3a 1087   = wceq 1541  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  Vcvv 3445  cin 3909  wss 3910  c0 4282  ifcif 4486   cuni 4865  cmpt 5188  dom cdm 5633  ran crn 5634  cres 5635  cima 5636  Rel wrel 5638  Ord word 6316  Oncon0 6317  Lim wlim 6318  suc csuc 6319   Fn wfn 6491  cfv 6496  recscrecs 8316  reccrdg 8355
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-ov 7360  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356
This theorem is referenced by:  dfrdg3  34371
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