Theorem fun2ssres 6537

Description: Equality of restrictions of a function and a subclass. (Contributed by NM, 16-Aug-1994.)

Assertion

Ref	Expression
fun2ssres	⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))

Proof of Theorem fun2ssres

Step	Hyp	Ref	Expression
1		resabs1 5965	. . . 4 ⊢ (𝐴 ⊆ dom 𝐺 → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐹 ↾ 𝐴))
2	1	eqcomd 2743	. . 3 ⊢ (𝐴 ⊆ dom 𝐺 → (𝐹 ↾ 𝐴) = ((𝐹 ↾ dom 𝐺) ↾ 𝐴))
3		funssres 6536	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
4	3	reseq1d 5937	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐺 ↾ 𝐴))
5	2, 4	sylan9eqr 2794	. 2 ⊢ (((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
6	5	3impa 1110	1 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ⊆ wss 3890  dom cdm 5624   ↾ cres 5626  Fun wfun 6486

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-res 5636  df-fun 6494

This theorem is referenced by:  frrlem10  8238  frrlem12  8240  fprresex  8253  tfrlem9  8317  tfrlem9a  8318  tfrlem11  8320  bnj1503  35007