Theorem bnj1502 35412

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 6894	. 2 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ∈ dom 𝐺) → (𝐹‘𝐴) = (𝐺‘𝐴))
5	1, 2, 3, 4	syl3anc 1398	1 ⊢ (𝜑 → (𝐹‘𝐴) = (𝐺‘𝐴))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145   ⊆ wss 3898  dom cdm 5647  Fun wfun 6521  ‘cfv 6527

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-pr 5390

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-res 5659  df-iota 6483  df-fun 6529  df-fv 6535

This theorem is used by:  bnj570  35469  bnj929  35500  bnj1450  35614  bnj1501  35631