Theorem bnj1502 34388

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1502.1	⊢ (𝜑 → Fun 𝐹)
bnj1502.2	⊢ (𝜑 → 𝐺 ⊆ 𝐹)
bnj1502.3	⊢ (𝜑 → 𝐴 ∈ dom 𝐺)

Assertion

Ref	Expression
bnj1502	⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))

Proof of Theorem bnj1502

Step	Hyp	Ref	Expression
1		bnj1502.1	. 2 ⊢ (𝜑 → Fun 𝐹)
2		bnj1502.2	. 2 ⊢ (𝜑 → 𝐺 ⊆ 𝐹)
3		bnj1502.3	. 2 ⊢ (𝜑 → 𝐴 ∈ dom 𝐺)
4		funssfv 6905	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴))
5	1, 2, 3, 4	syl3anc 1368	1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ⊆ wss 3943  dom cdm 5669  Fun wfun 6530  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6488  df-fun 6538  df-fv 6544

This theorem is referenced by:  bnj570  34445  bnj929  34476  bnj1450  34590  bnj1501  34607