Theorem f1ores 2813

Description: The restriction of a one-to-one function maps one-to-one onto the image.

Assertion

Ref	Expression
f1ores	|- ((F:A-1-1->B /\ C (_ A) -> (F |` C):C-1-1-onto->(F"C))

Proof of Theorem f1ores

Step	Hyp	Ref	Expression
1		fores 2794	. . . . 5 |- ((Fun F /\ C (_ dom F) -> (F |` C):C-onto->(F"C))
2		ffun 2754	. . . . . 6 |- (F:A-->B -> Fun F)
3	2	adantr 306	. . . . 5 |- ((F:A-->B /\ C (_ A) -> Fun F)
4		fdm 2756	. . . . . . 7 |- (F:A-->B -> dom F = A)
5	4	sseq2d 1528	. . . . . 6 |- (F:A-->B -> (C (_ dom F <-> C (_ A))
6	5	biimpar 325	. . . . 5 |- ((F:A-->B /\ C (_ A) -> C (_ dom F)
7	1, 3, 6	sylanc 361	. . . 4 |- ((F:A-->B /\ C (_ A) -> (F |` C):C-onto->(F"C))
8		funres11 2709	. . . 4 |- (Fun `'F -> Fun `'(F |` C))
9	7, 8	anim12i 268	. . 3 |- (((F:A-->B /\ C (_ A) /\ Fun `'F) -> ((F |` C):C-onto->(F"C) /\ Fun `'(F |` C)))
10	9	an1rs 373	. 2 |- (((F:A-->B /\ Fun `'F) /\ C (_ A) -> ((F |` C):C-onto->(F"C) /\ Fun `'(F |` C)))
11		df-f1 2435	. . 3 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
12	11	anbi1i 368	. 2 |- ((F:A-1-1->B /\ C (_ A) <-> ((F:A-->B /\ Fun `'F) /\ C (_ A))
13		f1o3 2805	. 2 |- ((F |` C):C-1-1-onto->(F"C) <-> ((F |` C):C-onto->(F"C) /\ Fun `'(F |` C)))
14	10, 12, 13	3imtr4 192	1 |- ((F:A-1-1->B /\ C (_ A) -> (F |` C):C-1-1-onto->(F"C))

Syntax hints:   -> wi 2   /\ wa 196   (_ wss 1487  `'ccnv 2409  dom cdm 2410   |` cres 2412  "cima 2413  Fun wfun 2416  -->wf 2418  -1-1->wf1 2419  -onto->wfo 2420  -1-1-onto->wf1o 2421

This theorem is referenced by:  f1imacnv 2814  f1imaen 3327  phplem5 3407  php3 3411  ssfi 3430  fiint 3445

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-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437

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