Theorem bnj1502 33854

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 6912	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴))
5	1, 2, 3, 4	syl3anc 1371	1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106   ⊆ wss 3948  dom cdm 5676  Fun wfun 6537  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-res 5688  df-iota 6495  df-fun 6545  df-fv 6551

This theorem is referenced by:  bnj570  33911  bnj929  33942  bnj1450  34056  bnj1501  34073