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Theorem bnj1502 34706
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 6914 . 2 ((Fun 𝐹𝐺𝐹𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
51, 2, 3, 4syl3anc 1368 1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  wss 3946  dom cdm 5674  Fun wfun 6540  cfv 6546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-br 5146  df-opab 5208  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-res 5686  df-iota 6498  df-fun 6548  df-fv 6554
This theorem is referenced by:  bnj570  34763  bnj929  34794  bnj1450  34908  bnj1501  34925
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