MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wfr2a Structured version   Visualization version   GIF version

Theorem wfr2a 7662
Description: A weak version of wfr2 7664 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 6402 . . 3 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2 predeq3 5891 . . . . 5 (𝑥 = 𝑋 → Pred(𝑅, 𝐴, 𝑥) = Pred(𝑅, 𝐴, 𝑋))
32reseq2d 5591 . . . 4 (𝑥 = 𝑋 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))
43fveq2d 6406 . . 3 (𝑥 = 𝑋 → (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
51, 4eqeq12d 2817 . 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 7656 . 2 (𝑥 ∈ dom 𝐹 → (𝐹𝑥) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥))))
105, 9vtoclga 3461 1 (𝑋 ∈ dom 𝐹 → (𝐹𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2155   Se wse 5262   We wwe 5263  dom cdm 5305  cres 5307  Predcpred 5886  cfv 6095  wrecscwrecs 7635
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2067  ax-7 2103  ax-8 2157  ax-9 2164  ax-10 2184  ax-11 2200  ax-12 2213  ax-13 2419  ax-ext 2781  ax-sep 4968  ax-nul 4977  ax-pow 5029  ax-pr 5090
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2060  df-eu 2633  df-mo 2634  df-clab 2789  df-cleq 2795  df-clel 2798  df-nfc 2933  df-ne 2975  df-ral 3097  df-rex 3098  df-reu 3099  df-rmo 3100  df-rab 3101  df-v 3389  df-sbc 3628  df-csb 3723  df-dif 3766  df-un 3768  df-in 3770  df-ss 3777  df-nul 4111  df-if 4274  df-sn 4365  df-pr 4367  df-op 4371  df-uni 4624  df-iun 4707  df-br 4838  df-opab 4900  df-mpt 4917  df-id 5213  df-po 5226  df-so 5227  df-fr 5264  df-se 5265  df-we 5266  df-xp 5311  df-rel 5312  df-cnv 5313  df-co 5314  df-dm 5315  df-rn 5316  df-res 5317  df-ima 5318  df-pred 5887  df-iota 6058  df-fun 6097  df-fn 6098  df-fv 6103  df-wrecs 7636
This theorem is referenced by:  wfr2  7664
  Copyright terms: Public domain W3C validator