Theorem wfr2aOLD 8340

Description: Obsolete version of wfr2a 8348 as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 30-Jul-2020.)

Hypotheses

Ref	Expression
wfr2aOLD.1	⊢ 𝑅 We 𝐴
wfr2aOLD.2	⊢ 𝑅 Se 𝐴
wfr2aOLD.3	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
wfr2aOLD	⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Proof of Theorem wfr2aOLD

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6876	. . 3 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
2		predeq3 6294	. . . . 5 ⊢ (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
3	2	reseq2d 5966	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))
4	3	fveq2d 6880	. . 3 ⊢ (𝑥 = 𝑋 → (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
5	1, 4	eqeq12d 2751	. 2 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑥) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))) ↔ (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))))
6		wfr2aOLD.1	. . 3 ⊢ 𝑅 We 𝐴
7		wfr2aOLD.2	. . 3 ⊢ 𝑅 Se 𝐴
8		wfr2aOLD.3	. . 3 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
9	6, 7, 8	wfrlem12OLD 8334	. 2 ⊢ (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))))
10	5, 9	vtoclga 3556	1 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   Se wse 5604   We wwe 5605  dom cdm 5654   ↾ cres 5656  Predcpred 6289  ‘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-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  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:  wfr2OLD  8342