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Theorem dfrdg2 31675
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 7493 . . 3 (𝑖 = 𝐼 → rec(𝐹, 𝑖) = rec(𝐹, 𝐼))
2 ifeq1 4081 . . . . . . . . 9 (𝑖 = 𝐼 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = if(𝑦 = ∅, 𝐼, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
32eqeq2d 2630 . . . . . . . 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 2983 . . . . . . 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 3048 . . . . 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 2739 . . . 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 4437 . . 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 2635 . 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 7491 . . 3 rec(𝐹, 𝑖) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝑖, if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))))))
11 dfrecs3 7454 . . 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 3198 . . . . . . . . . . . . 13 𝑓 ∈ V
1312resex 5431 . . . . . . . . . . . 12 (𝑓𝑦) ∈ V
14 eqeq1 2624 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓𝑦) → (𝑔 = ∅ ↔ (𝑓𝑦) = ∅))
15 relres 5414 . . . . . . . . . . . . . . . 16 Rel (𝑓𝑦)
16 reldm0 5332 . . . . . . . . . . . . . . . 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 5313 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑓𝑦) → dom 𝑔 = dom (𝑓𝑦))
20 limeq 5723 . . . . . . . . . . . . . . . 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 5340 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑓𝑦) → ran 𝑔 = ran (𝑓𝑦))
23 df-ima 5117 . . . . . . . . . . . . . . . . 17 (𝑓𝑦) = ran (𝑓𝑦)
2422, 23syl6eqr 2672 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑓𝑦) → ran 𝑔 = (𝑓𝑦))
2524unieqd 4437 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓𝑦) → ran 𝑔 = (𝑓𝑦))
26 id 22 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑓𝑦) → 𝑔 = (𝑓𝑦))
2719unieqd 4437 . . . . . . . . . . . . . . . . 17 (𝑔 = (𝑓𝑦) → dom 𝑔 = dom (𝑓𝑦))
2826, 27fveq12d 6184 . . . . . . . . . . . . . . . 16 (𝑔 = (𝑓𝑦) → (𝑔 dom 𝑔) = ((𝑓𝑦)‘ dom (𝑓𝑦)))
2928fveq2d 6182 . . . . . . . . . . . . . . 15 (𝑔 = (𝑓𝑦) → (𝐹‘(𝑔 dom 𝑔)) = (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))))
3021, 25, 29ifbieq12d 4104 . . . . . . . . . . . . . 14 (𝑔 = (𝑓𝑦) → if(Lim dom 𝑔, ran 𝑔, (𝐹‘(𝑔 dom 𝑔))) = if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦)))))
3118, 30ifbieq2d 4102 . . . . . . . . . . . . 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 2620 . . . . . . . . . . . . 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 3198 . . . . . . . . . . . . . 14 𝑖 ∈ V
34 imaexg 7088 . . . . . . . . . . . . . . . . 17 (𝑓 ∈ V → (𝑓𝑦) ∈ V)
3512, 34ax-mp 5 . . . . . . . . . . . . . . . 16 (𝑓𝑦) ∈ V
3635uniex 6938 . . . . . . . . . . . . . . 15 (𝑓𝑦) ∈ V
37 fvex 6188 . . . . . . . . . . . . . . 15 (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))) ∈ V
3836, 37ifex 4147 . . . . . . . . . . . . . 14 if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦)))) ∈ V
3933, 38ifex 4147 . . . . . . . . . . . . 13 if(dom (𝑓𝑦) = ∅, 𝑖, if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))))) ∈ V
4031, 32, 39fvmpt 6269 . . . . . . . . . . . 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 5407 . . . . . . . . . . . . 13 dom (𝑓𝑦) = (𝑦 ∩ dom 𝑓)
43 onelss 5754 . . . . . . . . . . . . . . . . 17 (𝑥 ∈ On → (𝑦𝑥𝑦𝑥))
4443imp 445 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦𝑥)
45443adant2 1078 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → 𝑦𝑥)
46 fndm 5978 . . . . . . . . . . . . . . . 16 (𝑓 Fn 𝑥 → dom 𝑓 = 𝑥)
47463ad2ant2 1081 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → dom 𝑓 = 𝑥)
4845, 47sseqtr4d 3634 . . . . . . . . . . . . . 14 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → 𝑦 ⊆ dom 𝑓)
49 df-ss 3581 . . . . . . . . . . . . . 14 (𝑦 ⊆ dom 𝑓 ↔ (𝑦 ∩ dom 𝑓) = 𝑦)
5048, 49sylib 208 . . . . . . . . . . . . 13 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → (𝑦 ∩ dom 𝑓) = 𝑦)
5142, 50syl5eq 2666 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑓 Fn 𝑥𝑦𝑥) → dom (𝑓𝑦) = 𝑦)
52 eqeq1 2624 . . . . . . . . . . . . . 14 (dom (𝑓𝑦) = 𝑦 → (dom (𝑓𝑦) = ∅ ↔ 𝑦 = ∅))
53 limeq 5723 . . . . . . . . . . . . . . 15 (dom (𝑓𝑦) = 𝑦 → (Lim dom (𝑓𝑦) ↔ Lim 𝑦))
54 unieq 4435 . . . . . . . . . . . . . . . . 17 (dom (𝑓𝑦) = 𝑦 dom (𝑓𝑦) = 𝑦)
5554fveq2d 6182 . . . . . . . . . . . . . . . 16 (dom (𝑓𝑦) = 𝑦 → ((𝑓𝑦)‘ dom (𝑓𝑦)) = ((𝑓𝑦)‘ 𝑦))
5655fveq2d 6182 . . . . . . . . . . . . . . 15 (dom (𝑓𝑦) = 𝑦 → (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦))) = (𝐹‘((𝑓𝑦)‘ 𝑦)))
5753, 56ifbieq2d 4102 . . . . . . . . . . . . . 14 (dom (𝑓𝑦) = 𝑦 → if(Lim dom (𝑓𝑦), (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ dom (𝑓𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))))
5852, 57ifbieq2d 4102 . . . . . . . . . . . . 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 5736 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ On ∧ 𝑦𝑥) → 𝑦 ∈ On)
60 eloni 5721 . . . . . . . . . . . . . . . 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 7030 . . . . . . . . . . . . . . 15 (Ord 𝑦 ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ Lim 𝑦))
64 iftrue 4083 . . . . . . . . . . . . . . . . 17 (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = 𝑖)
65 iftrue 4083 . . . . . . . . . . . . . . . . 17 (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = 𝑖)
6664, 65eqtr4d 2657 . . . . . . . . . . . . . . . 16 (𝑦 = ∅ → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))
67 vex 3198 . . . . . . . . . . . . . . . . . . . . . . 23 𝑧 ∈ V
6867sucid 5792 . . . . . . . . . . . . . . . . . . . . . 22 𝑧 ∈ suc 𝑧
69 fvres 6194 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ suc 𝑧 → ((𝑓 ↾ suc 𝑧)‘𝑧) = (𝑓𝑧))
7068, 69ax-mp 5 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓 ↾ suc 𝑧)‘𝑧) = (𝑓𝑧)
71 eloni 5721 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ On → Ord 𝑧)
72 ordunisuc 7017 . . . . . . . . . . . . . . . . . . . . . . 23 (Ord 𝑧 suc 𝑧 = 𝑧)
7371, 72syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ On → suc 𝑧 = 𝑧)
7473fveq2d 6182 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘ suc 𝑧) = ((𝑓 ↾ suc 𝑧)‘𝑧))
7573fveq2d 6182 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ On → (𝑓 suc 𝑧) = (𝑓𝑧))
7670, 74, 753eqtr4a 2680 . . . . . . . . . . . . . . . . . . . 20 (𝑧 ∈ On → ((𝑓 ↾ suc 𝑧)‘ suc 𝑧) = (𝑓 suc 𝑧))
7776fveq2d 6182 . . . . . . . . . . . . . . . . . . 19 (𝑧 ∈ On → (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)) = (𝐹‘(𝑓 suc 𝑧)))
78 nsuceq0 5793 . . . . . . . . . . . . . . . . . . . . . 22 suc 𝑧 ≠ ∅
7978neii 2793 . . . . . . . . . . . . . . . . . . . . 21 ¬ suc 𝑧 = ∅
8079iffalsei 4087 . . . . . . . . . . . . . . . . . . . 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 7027 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ V → ¬ Lim suc 𝑧)
82 iffalse 4086 . . . . . . . . . . . . . . . . . . . . 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 2642 . . . . . . . . . . . . . . . . . . 19 if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)))) = (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧))
8579iffalsei 4087 . . . . . . . . . . . . . . . . . . . 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 4086 . . . . . . . . . . . . . . . . . . . . 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 2642 . . . . . . . . . . . . . . . . . . 19 if(suc 𝑧 = ∅, 𝑖, if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))) = (𝐹‘(𝑓 suc 𝑧))
8977, 84, 883eqtr4g 2679 . . . . . . . . . . . . . . . . . 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 2624 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
91 limeq 5723 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = suc 𝑧 → (Lim 𝑦 ↔ Lim suc 𝑧))
92 reseq2 5380 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = suc 𝑧 → (𝑓𝑦) = (𝑓 ↾ suc 𝑧))
93 unieq 4435 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 = suc 𝑧 𝑦 = suc 𝑧)
9492, 93fveq12d 6184 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = suc 𝑧 → ((𝑓𝑦)‘ 𝑦) = ((𝑓 ↾ suc 𝑧)‘ suc 𝑧))
9594fveq2d 6182 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = suc 𝑧 → (𝐹‘((𝑓𝑦)‘ 𝑦)) = (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧)))
9691, 95ifbieq2d 4102 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = suc 𝑧 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))) = if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘((𝑓 ↾ suc 𝑧)‘ suc 𝑧))))
9790, 96ifbieq2d 4102 . . . . . . . . . . . . . . . . . . 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 6182 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = suc 𝑧 → (𝑓 𝑦) = (𝑓 suc 𝑧))
9998fveq2d 6182 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = suc 𝑧 → (𝐹‘(𝑓 𝑦)) = (𝐹‘(𝑓 suc 𝑧)))
10091, 99ifbieq2d 4102 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = suc 𝑧 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))) = if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧))))
10190, 100ifbieq2d 4102 . . . . . . . . . . . . . . . . . . 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 2635 . . . . . . . . . . . . . . . . . 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 3023 . . . . . . . . . . . . . . . 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 4083 . . . . . . . . . . . . . . . . . 18 (Lim 𝑦 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))) = (𝑓𝑦))
106 df-lim 5716 . . . . . . . . . . . . . . . . . . . . 21 (Lim 𝑦 ↔ (Ord 𝑦𝑦 ≠ ∅ ∧ 𝑦 = 𝑦))
107106simp2bi 1075 . . . . . . . . . . . . . . . . . . . 20 (Lim 𝑦𝑦 ≠ ∅)
108107neneqd 2796 . . . . . . . . . . . . . . . . . . 19 (Lim 𝑦 → ¬ 𝑦 = ∅)
109108iffalsed 4088 . . . . . . . . . . . . . . . . . 18 (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦))))
110 iftrue 4083 . . . . . . . . . . . . . . . . . 18 (Lim 𝑦 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))) = (𝑓𝑦))
111105, 109, 1103eqtr4d 2664 . . . . . . . . . . . . . . . . 17 (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘((𝑓𝑦)‘ 𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))
112108iffalsed 4088 . . . . . . . . . . . . . . . . 17 (Lim 𝑦 → if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))
113111, 112eqtr4d 2657 . . . . . . . . . . . . . . . 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 1389 . . . . . . . . . . . . . . 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 2676 . . . . . . . . . . . 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 2666 . . . . . . . . . 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 2630 . . . . . . . . 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 1263 . . . . . . . 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 2982 . . . . . . 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 3036 . . . . 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 2737 . . . 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 4436 . . 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 2646 . 2 rec(𝐹, 𝑖) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, 𝑖, if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))}
1289, 127vtoclg 3261 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 1481  wcel 1988  {cab 2606  wne 2791  wral 2909  wrex 2910  Vcvv 3195  cin 3566  wss 3567  c0 3907  ifcif 4077   cuni 4427  cmpt 4720  dom cdm 5104  ran crn 5105  cres 5106  cima 5107  Rel wrel 5109  Ord word 5710  Oncon0 5711  Lim wlim 5712  suc csuc 5713   Fn wfn 5871  cfv 5876  recscrecs 7452  reccrdg 7490
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-fv 5884  df-wrecs 7392  df-recs 7453  df-rdg 7491
This theorem is referenced by:  dfrdg3  31676
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