Theorem fun2ssres 2699

Description: Equality of restrictions of a function and a subclass.

Assertion

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

Proof of Theorem fun2ssres

Step	Hyp	Ref	Expression
1		funssres 2698	. . . . 5 ⊢ ((Fun F ∧ G ⊆ F) → (F ↾ dom G) = G)
2		reseq1 2575	. . . . 5 ⊢ ((F ↾ dom G) = G → ((F ↾ dom G) ↾ A) = (G ↾ A))
3	1, 2	syl 12	. . . 4 ⊢ ((Fun F ∧ G ⊆ F) → ((F ↾ dom G) ↾ A) = (G ↾ A))
4	3	cleqcomd 1106	. . 3 ⊢ ((Fun F ∧ G ⊆ F) → (G ↾ A) = ((F ↾ dom G) ↾ A))
5		resabs1 2592	. . 3 ⊢ (A ⊆ dom G → ((F ↾ dom G) ↾ A) = (F ↾ A))
6	4, 5	sylan9eq 1144	. 2 ⊢ (((Fun F ∧ G ⊆ F) ∧ A ⊆ dom G) → (G ↾ A) = (F ↾ A))
7	6	cleqcomd 1106	1 ⊢ (((Fun F ∧ G ⊆ F) ∧ A ⊆ dom G) → (F ↾ A) = (G ↾ A))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ⊆ wss 1487  dom cdm 2410   ↾ cres 2412  Fun wfun 2416

This theorem is referenced by:  tfrlem9 2957  tfrlem11 2959

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-res 2430  df-fun 2432

metamath.org