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Theorem bnj1502 32822
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 6790 . 2 ((Fun 𝐹𝐺𝐹𝐴 ∈ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
51, 2, 3, 4syl3anc 1370 1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2110  wss 3892  dom cdm 5589  Fun wfun 6425  cfv 6431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rex 3072  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-br 5080  df-opab 5142  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-iota 6389  df-fun 6433  df-fv 6439
This theorem is referenced by:  bnj570  32879  bnj929  32910  bnj1450  33024  bnj1501  33041
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