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Theorem wfr2a 7392
Description: A weak version of wfr2 7394 which is useful for proofs that avoid the Axiom of Replacement. (Contributed by Scott Fenton, 30-Jul-2020.)
Hypotheses
Ref Expression
wfr2a.1 𝑅 We 𝐴
wfr2a.2 𝑅 Se 𝐴
wfr2a.3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
Assertion
Ref Expression
wfr2a (𝑋 ∈ dom 𝐹 → (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Proof of Theorem wfr2a
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6158 . . 3 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2 predeq3 5653 . . . . 5 (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
32reseq2d 5366 . . . 4 (𝑥 = 𝑋 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))
43fveq2d 6162 . . 3 (𝑥 = 𝑋 → (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
51, 4eqeq12d 2636 . 2 (𝑥 = 𝑋 → ((𝐹𝑥) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))) ↔ (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))))
6 wfr2a.1 . . 3 𝑅 We 𝐴
7 wfr2a.2 . . 3 𝑅 Se 𝐴
8 wfr2a.3 . . 3 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
96, 7, 8wfrlem12 7386 . 2 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))))
105, 9vtoclga 3262 1 (𝑋 ∈ dom 𝐹 → (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987   Se wse 5041   We wwe 5042  dom cdm 5084  cres 5086  Predcpred 5648  cfv 5857  wrecscwrecs 7366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-se 5044  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-iota 5820  df-fun 5859  df-fn 5860  df-fv 5865  df-wrecs 7367
This theorem is referenced by:  wfr2  7394
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