Theorem fun2ssres 6594

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 6012	. . . 4 ⊢ (𝐴 ⊆ dom 𝐺 → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐹 ↾ 𝐴))
2	1	eqcomd 2739	. . 3 ⊢ (𝐴 ⊆ dom 𝐺 → (𝐹 ↾ 𝐴) = ((𝐹 ↾ dom 𝐺) ↾ 𝐴))
3		funssres 6593	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
4	3	reseq1d 5981	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐺 ↾ 𝐴))
5	2, 4	sylan9eqr 2795	. 2 ⊢ (((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
6	5	3impa 1111	1 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ⊆ wss 3949  dom cdm 5677   ↾ cres 5679  Fun wfun 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-fun 6546

This theorem is referenced by:  frrlem10  8280  frrlem12  8282  fprresex  8295  wfrlem12OLD  8320  wfrlem14OLD  8322  wfrlem17OLD  8325  tfrlem9  8385  tfrlem9a  8386  tfrlem11  8388  bnj1503  33860