Theorem fores 2794

Description: Restriction of a function.

Assertion

Ref	Expression
fores	⊢ ((Fun F ∧ A ⊆ dom F) → (F ↾ A):A–onto→(F “ A))

Proof of Theorem fores

Step	Hyp	Ref	Expression
1		funres 2697	. . 3 ⊢ (Fun F → Fun (F ↾ A))
2	1	anim1i 269	. 2 ⊢ ((Fun F ∧ A ⊆ dom F) → (Fun (F ↾ A) ∧ A ⊆ dom F))
3		df-fn 2433	. . 3 ⊢ ((F ↾ A) Fn A ↔ (Fun (F ↾ A) ∧ dom (F ↾ A) = A))
4		df-fo 2436	. . . 4 ⊢ ((F ↾ A):A–onto→(F “ A) ↔ ((F ↾ A) Fn A ∧ ran (F ↾ A) = (F “ A)))
5		df-ima 2431	. . . . 5 ⊢ (F “ A) = ran (F ↾ A)
6	5	cleqcomi 1105	. . . 4 ⊢ ran (F ↾ A) = (F “ A)
7	4, 6	mpbiranr 548	. . 3 ⊢ ((F ↾ A):A–onto→(F “ A) ↔ (F ↾ A) Fn A)
8		ssdmres 2585	. . . 4 ⊢ (A ⊆ dom F ↔ dom (F ↾ A) = A)
9	8	anbi2i 367	. . 3 ⊢ ((Fun (F ↾ A) ∧ A ⊆ dom F) ↔ (Fun (F ↾ A) ∧ dom (F ↾ A) = A))
10	3, 7, 9	3bitr4 158	. 2 ⊢ ((F ↾ A):A–onto→(F “ A) ↔ (Fun (F ↾ A) ∧ A ⊆ dom F))
11	2, 10	sylibr 175	1 ⊢ ((Fun F ∧ A ⊆ dom F) → (F ↾ A):A–onto→(F “ A))

Syntax hints:   → wi 2   ∧ wa 196   = wceq 1091   ⊆ wss 1487  dom cdm 2410  ran crn 2411   ↾ cres 2412   “ cima 2413  Fun wfun 2416   Fn wfn 2417  –onto→wfo 2420

This theorem is referenced by:  f1ores 2813  f1oweOLD 2944

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-ima 2431  df-fun 2432  df-fn 2433  df-fo 2436

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