Theorem bnj1502 31435

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 6432	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴))
5	1, 2, 3, 4	syl3anc 1491	1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))

Syntax hints:   → wi 4   = wceq 1653   ∈ wcel 2157   ⊆ wss 3769  dom cdm 5312  Fun wfun 6095  ‘cfv 6101

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-res 5324  df-iota 6064  df-fun 6103  df-fv 6109

This theorem is referenced by:  bnj570  31492  bnj929  31523  bnj1450  31635  bnj1501  31652