Theorem bnj1503 32829

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	⊢ (𝜑 → 𝐺 ⊆ 𝐹)
bnj1503.3	⊢ (𝜑 → 𝐴 ⊆ dom 𝐺)

Assertion

Ref	Expression
bnj1503	⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))

Proof of Theorem bnj1503

Step	Hyp	Ref	Expression
1		bnj1503.1	. 2 ⊢ (𝜑 → Fun 𝐹)
2		bnj1503.2	. 2 ⊢ (𝜑 → 𝐺 ⊆ 𝐹)
3		bnj1503.3	. 2 ⊢ (𝜑 → 𝐴 ⊆ dom 𝐺)
4		fun2ssres 6479	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
5	1, 2, 3, 4	syl3anc 1370	1 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1539   ⊆ wss 3887  dom cdm 5589   ↾ cres 5591  Fun wfun 6427

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 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-res 5601  df-fun 6435

This theorem is referenced by:  bnj1442  33029  bnj1450  33030  bnj1501  33047