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Theorem wfr2aOLD 8365
Description: Obsolete version of wfr2a 8373 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 → (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Proof of Theorem wfr2aOLD
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6907 . . 3 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2 predeq3 6327 . . . . 5 (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
32reseq2d 6000 . . . 4 (𝑥 = 𝑋 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))
43fveq2d 6911 . . 3 (𝑥 = 𝑋 → (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
51, 4eqeq12d 2751 . 2 (𝑥 = 𝑋 → ((𝐹𝑥) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))) ↔ (𝐹𝑋) = (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)
96, 7, 8wfrlem12OLD 8359 . 2 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))))
105, 9vtoclga 3577 1 (𝑋 ∈ dom 𝐹 → (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106   Se wse 5639   We wwe 5640  dom cdm 5689  cres 5691  Predcpred 6322  cfv 6563  wrecscwrecs 8335
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-ov 7434  df-2nd 8014  df-frecs 8305  df-wrecs 8336
This theorem is referenced by:  wfr2OLD  8367
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