Theorem fun2ssres 6587

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 6005	. . . 4 ⊢ (𝐴 ⊆ dom 𝐺 → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐹 ↾ 𝐴))
2	1	eqcomd 2732	. . 3 ⊢ (𝐴 ⊆ dom 𝐺 → (𝐹 ↾ 𝐴) = ((𝐹 ↾ dom 𝐺) ↾ 𝐴))
3		funssres 6586	. . . 4 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → (𝐹 ↾ dom 𝐺) = 𝐺)
4	3	reseq1d 5974	. . 3 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) → ((𝐹 ↾ dom 𝐺) ↾ 𝐴) = (𝐺 ↾ 𝐴))
5	2, 4	sylan9eqr 2788	. 2 ⊢ (((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹) ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))
6	5	3impa 1107	1 ⊢ ((Fun 𝐹 ∧ 𝐺 ⊆ 𝐹 ∧ 𝐴 ⊆ dom 𝐺) → (𝐹 ↾ 𝐴) = (𝐺 ↾ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ⊆ wss 3943  dom cdm 5669   ↾ cres 5671  Fun wfun 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-fun 6539

This theorem is referenced by:  frrlem10  8281  frrlem12  8283  fprresex  8296  wfrlem12OLD  8321  wfrlem14OLD  8323  wfrlem17OLD  8326  tfrlem9  8386  tfrlem9a  8387  tfrlem11  8389  bnj1503  34389