Theorem bnj1503 30173

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 5845	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
5	1, 2, 3, 4	syl3anc 1318	1 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))

Syntax hints:   → wi 4   = wceq 1475   ⊆ wss 3540  dom cdm 5038   ↾ cres 5040  Fun wfun 5798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-fun 5806

This theorem is referenced by:  bnj1442  30371  bnj1450  30372  bnj1501  30389