Theorem bnj1503 35345

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 6582	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
5	1, 2, 3, 4	syl3anc 1398	1 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3902  dom cdm 5659   ↾ cres 5661  Fun wfun 6531

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-res 5671  df-fun 6539

This theorem is used by:  bnj1442  35545  bnj1450  35546  bnj1501  35563