Theorem fnssres 2734

Description: Restriction of a function with a subclass of its domain.

Assertion

Ref	Expression
fnssres	⊢ ((F Fn A ∧ B ⊆ A) → (F ↾ B) Fn B)

Proof of Theorem fnssres

Step	Hyp	Ref	Expression
1		fnfun 2721	. . . . 5 ⊢ (F Fn A → Fun F)
2		funres 2697	. . . . 5 ⊢ (Fun F → Fun (F ↾ B))
3	1, 2	syl 12	. . . 4 ⊢ (F Fn A → Fun (F ↾ B))
4	3	adantr 306	. . 3 ⊢ ((F Fn A ∧ B ⊆ A) → Fun (F ↾ B))
5		fndm 2723	. . . . . 6 ⊢ (F Fn A → dom F = A)
6	5	sseq2d 1528	. . . . 5 ⊢ (F Fn A → (B ⊆ dom F ↔ B ⊆ A))
7	6	biimpar 325	. . . 4 ⊢ ((F Fn A ∧ B ⊆ A) → B ⊆ dom F)
8		ssdmres 2585	. . . 4 ⊢ (B ⊆ dom F ↔ dom (F ↾ B) = B)
9	7, 8	sylib 173	. . 3 ⊢ ((F Fn A ∧ B ⊆ A) → dom (F ↾ B) = B)
10	4, 9	jca 236	. 2 ⊢ ((F Fn A ∧ B ⊆ A) → (Fun (F ↾ B) ∧ dom (F ↾ B) = B))
11		df-fn 2433	. 2 ⊢ ((F ↾ B) Fn B ↔ (Fun (F ↾ B) ∧ dom (F ↾ B) = B))
12	10, 11	sylibr 175	1 ⊢ ((F Fn A ∧ B ⊆ A) → (F ↾ B) Fn B)

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ⊆ wss 1487  dom cdm 2410   ↾ cres 2412  Fun wfun 2416   Fn wfn 2417

This theorem is referenced by:  fnresin1 2735  fnresin2 2736  fssres 2764  fvreseq 2882  fnressn 2897  tz7.48lem 2993  tz7.49c 2998  df1st2 3098

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  df-fn 2433

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