Theorem bnj1502 34821

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	⊢ (𝜑 → 𝐺 ⊆ 𝐹)
bnj1502.3	⊢ (𝜑 → 𝐴 ∈ dom 𝐺)

Assertion

Ref	Expression
bnj1502	⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))

Proof of Theorem bnj1502

Step	Hyp	Ref	Expression
1		bnj1502.1	. 2 ⊢ (𝜑 → Fun 𝐹)
2		bnj1502.2	. 2 ⊢ (𝜑 → 𝐺 ⊆ 𝐹)
3		bnj1502.3	. 2 ⊢ (𝜑 → 𝐴 ∈ dom 𝐺)
4		funssfv 6907	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴))
5	1, 2, 3, 4	syl3anc 1372	1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107   ⊆ wss 3931  dom cdm 5665  Fun wfun 6535  ‘cfv 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-res 5677  df-iota 6494  df-fun 6543  df-fv 6549

This theorem is referenced by:  bnj570  34878  bnj929  34909  bnj1450  35023  bnj1501  35040