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Theorem wfr2a 7815
Description: A weak version of wfr2 7817 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 6530 . . 3 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2 predeq3 6019 . . . . 5 (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
32reseq2d 5726 . . . 4 (𝑥 = 𝑋 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))
43fveq2d 6534 . . 3 (𝑥 = 𝑋 → (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
51, 4eqeq12d 2808 . 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 7809 . 2 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))))
105, 9vtoclga 3512 1 (𝑋 ∈ dom 𝐹 → (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1520  wcel 2079   Se wse 5392   We wwe 5393  dom cdm 5435  cres 5437  Predcpred 6014  cfv 6217  wrecscwrecs 7788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pow 5150  ax-pr 5214
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1079  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-reu 3110  df-rmo 3111  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-iun 4821  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-po 5354  df-so 5355  df-fr 5394  df-se 5395  df-we 5396  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-pred 6015  df-iota 6181  df-fun 6219  df-fn 6220  df-fv 6225  df-wrecs 7789
This theorem is referenced by:  wfr2  7817
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