Theorem bnj1503 34842

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 6613	. 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 1537   ⊆ wss 3963  dom cdm 5689   ↾ cres 5691  Fun wfun 6557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-fun 6565

This theorem is referenced by:  bnj1442  35042  bnj1450  35043  bnj1501  35060