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Theorem bnj1503 31045
 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 5969 . 2 ((Fun 𝐹𝐺𝐹𝐴 ⊆ dom 𝐺) → (𝐹𝐴) = (𝐺𝐴))
51, 2, 3, 4syl3anc 1366 1 (𝜑 → (𝐹𝐴) = (𝐺𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1523   ⊆ wss 3607  dom cdm 5143   ↾ cres 5145  Fun wfun 5920 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-fun 5928 This theorem is referenced by:  bnj1442  31243  bnj1450  31244  bnj1501  31261
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