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Theorem bnj1503 34863
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1503.1 (𝜑 → Fun 𝐹)
bnj1503.2 (𝜑𝐺𝐹)
bnj1503.3 (𝜑𝐴 ⊆ dom 𝐺)
Assertion
Ref Expression
bnj1503 (𝜑 → (𝐹𝐴) = (𝐺𝐴))

Proof of Theorem bnj1503
StepHypRef Expression
1 bnj1503.1 . 2 (𝜑 → Fun 𝐹)
2 bnj1503.2 . 2 (𝜑𝐺𝐹)
3 bnj1503.3 . 2 (𝜑𝐴 ⊆ dom 𝐺)
4 fun2ssres 6611 . 2 ((Fun 𝐹𝐺𝐹𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
51, 2, 3, 4syl3anc 1373 1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wss 3951  dom cdm 5685  cres 5687  Fun wfun 6555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-fun 6563
This theorem is referenced by:  bnj1442  35063  bnj1450  35064  bnj1501  35081
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