Theorem wrecseq123 7408

Description: General equality theorem for the well-founded recursive function generator. (Contributed by Scott Fenton, 7-Jun-2018.)

Assertion

Ref	Expression
wrecseq123	⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))

Proof of Theorem wrecseq123

Dummy variables 𝑓 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 3627	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))
2	1	3ad2ant2 1083	. . . . . . 7 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))
3		predeq1 5682	. . . . . . . . . . 11 ⊢ (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐴, 𝑦))
4		predeq2 5683	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → Pred(𝑆, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
5	3, 4	sylan9eq 2676	. . . . . . . . . 10 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
6	5	3adant3 1081	. . . . . . . . 9 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
7	6	sseq1d 3632	. . . . . . . 8 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥))
8	7	ralbidv 2986	. . . . . . 7 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ ∀𝑦 ∈ 𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥))
9	2, 8	anbi12d 747	. . . . . 6 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → ((𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ↔ (𝑥 ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥)))
10		simp3 1063	. . . . . . . . 9 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → 𝐹 = 𝐺)
11	5	reseq2d 5396	. . . . . . . . . 10 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))
12	11	3adant3 1081	. . . . . . . . 9 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))
13	10, 12	fveq12d 6197	. . . . . . . 8 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))
14	13	eqeq2d 2632	. . . . . . 7 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))))
15	14	ralbidv 2986	. . . . . 6 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))))
16	9, 15	3anbi23d 1402	. . . . 5 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → ((𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))))
17	16	exbidv 1850	. . . 4 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))))
18	17	abbidv 2741	. . 3 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))})
19	18	unieqd 4446	. 2 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))})
20		df-wrecs 7407	. 2 ⊢ wrecs(𝑅, 𝐴, 𝐹) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
21		df-wrecs 7407	. 2 ⊢ wrecs(𝑆, 𝐵, 𝐺) = ∪ {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐵 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))}
22	19, 20, 21	3eqtr4g 2681	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704  {cab 2608  ∀wral 2912   ⊆ wss 3574  ∪ cuni 4436   ↾ cres 5116  Predcpred 5679   Fn wfn 5883  ‘cfv 5888  wrecscwrecs 7406

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fv 5896  df-wrecs 7407

This theorem is referenced by:  wrecseq1  7410  wrecseq2  7411  wrecseq3  7412