Theorem wrecseq123OLD 8314

Description: Obsolete version of wrecseq123 8313 as of 17-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 7-Jun-2018.)

Assertion

Ref	Expression
wrecseq123OLD	⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))

Proof of Theorem wrecseq123OLD

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

Step	Hyp	Ref	Expression
1		sseq2 3985	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))
2	1	3ad2ant2 1134	. . . . . . 7 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (𝑥 ⊆ 𝐴 ↔ 𝑥 ⊆ 𝐵))
3		predeq1 6292	. . . . . . . . . . 11 ⊢ (𝑅 = 𝑆 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐴, 𝑦))
4		predeq2 6293	. . . . . . . . . . 11 ⊢ (𝐴 = 𝐵 → Pred(𝑆, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
5	3, 4	sylan9eq 2790	. . . . . . . . . 10 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
6	5	3adant3 1132	. . . . . . . . 9 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑆, 𝐵, 𝑦))
7	6	sseq1d 3990	. . . . . . . 8 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ↔ Pred(𝑆, 𝐵, 𝑦) ⊆ 𝑥))
8	7	ralbidv 3163	. . . . . . 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 632	. . . . . 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 1138	. . . . . . . . 9 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → 𝐹 = 𝐺)
11	5	reseq2d 5966	. . . . . . . . . 10 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))
12	11	3adant3 1132	. . . . . . . . 9 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝑓 ↾ Pred(𝑆, 𝐵, 𝑦)))
13	10, 12	fveq12d 6883	. . . . . . . 8 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐺‘(𝑓 ↾ Pred(𝑆, 𝐵, 𝑦))))
14	13	eqeq2d 2746	. . . . . . 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 3163	. . . . . 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 1441	. . . . 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 1921	. . . 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 2801	. . 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 4896	. 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		dfwrecsOLD 8312	. 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		dfwrecsOLD 8312	. 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 2795	1 ⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵 ∧ 𝐹 = 𝐺) → wrecs(𝑅, 𝐴, 𝐹) = wrecs(𝑆, 𝐵, 𝐺))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779  {cab 2713  ∀wral 3051   ⊆ wss 3926  ∪ cuni 4883   ↾ cres 5656  Predcpred 6289   Fn wfn 6526  ‘cfv 6531  wrecscwrecs 8310

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-fo 6537  df-fv 6539  df-ov 7408  df-2nd 7989  df-frecs 8280  df-wrecs 8311

This theorem is referenced by: (None)