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Theorem bnj1502 30892
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1502.1 (𝜑 → Fun 𝐹)
bnj1502.2 (𝜑𝐺𝐹)
bnj1502.3 (𝜑𝐴 ∈ dom 𝐺)
Assertion
Ref Expression
bnj1502 (𝜑 → (𝐹𝐴) = (𝐺𝐴))

Proof of Theorem bnj1502
StepHypRef Expression
1 bnj1502.1 . 2 (𝜑 → Fun 𝐹)
2 bnj1502.2 . 2 (𝜑𝐺𝐹)
3 bnj1502.3 . 2 (𝜑𝐴 ∈ dom 𝐺)
4 funssfv 6196 . 2 ((Fun 𝐹𝐺𝐹𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
51, 2, 3, 4syl3anc 1324 1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1481  wcel 1988  wss 3567  dom cdm 5104  Fun wfun 5870  cfv 5876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-res 5116  df-iota 5839  df-fun 5878  df-fv 5884
This theorem is referenced by:  bnj570  30949  bnj929  30980  bnj1450  31092  bnj1501  31109
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