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Theorem bnj1502 34484
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 6921 . 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 1533  wcel 2098  wss 3947  dom cdm 5680  Fun wfun 6545  cfv 6551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-res 5692  df-iota 6503  df-fun 6553  df-fv 6559
This theorem is referenced by:  bnj570  34541  bnj929  34572  bnj1450  34686  bnj1501  34703
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