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Theorem bnj1503 32816
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 6473 . 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 1539  wss 3888  dom cdm 5586  cres 5588  Fun wfun 6422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3433  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4259  df-if 4462  df-sn 4564  df-pr 4566  df-op 4570  df-br 5076  df-opab 5138  df-id 5486  df-xp 5592  df-rel 5593  df-cnv 5594  df-co 5595  df-dm 5596  df-res 5598  df-fun 6430
This theorem is referenced by:  bnj1442  33016  bnj1450  33017  bnj1501  33034
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