Theorem bnj1503 35546

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 6602	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
5	1, 2, 3, 4	syl3anc 1399	1 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ⊆ wss 3900  dom cdm 5655   ↾ cres 5657  Fun wfun 6549

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-res 5667  df-fun 6557

This theorem is used by:  bnj1442  35746  bnj1450  35747  bnj1501  35764