Theorem f1ores 3703

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 3681	. . . . 5 |- ((Fun F /\ C (_ dom F) -> (F |` C):C-onto->(F"C))
2		ffun 3629	. . . . . 6 |- (F:A-->B -> Fun F)
3	2	adantr 389	. . . . 5 |- ((F:A-->B /\ C (_ A) -> Fun F)
4		fdm 3631	. . . . . . 7 |- (F:A-->B -> dom F = A)
5	4	sseq2d 2089	. . . . . 6 |- (F:A-->B -> (C (_ dom F <-> C (_ A))
6	5	biimpar 417	. . . . 5 |- ((F:A-->B /\ C (_ A) -> C (_ dom F)
7	1, 3, 6	sylanc 471	. . . 4 |- ((F:A-->B /\ C (_ A) -> (F |` C):C-onto->(F"C))
8		funres11 3567	. . . 4 |- (Fun `'F -> Fun `'(F |` C))
9	7, 8	anim12i 333	. . 3 |- (((F:A-->B /\ C (_ A) /\ Fun `'F) -> ((F |` C):C-onto->(F"C) /\ Fun `'(F |` C)))
10	9	an1rs 489	. 2 |- (((F:A-->B /\ Fun `'F) /\ C (_ A) -> ((F |` C):C-onto->(F"C) /\ Fun `'(F |` C)))
11		df-f1 3195	. . 3 |- (F:A-1-1->B <-> (F:A-->B /\ Fun `'F))
12	11	anbi1i 481	. 2 |- ((F:A-1-1->B /\ C (_ A) <-> ((F:A-->B /\ Fun `'F) /\ C (_ A))
13		f1o3 3694	. 2 |- ((F |` C):C-1-1-onto->(F"C) <-> ((F |` C):C-onto->(F"C) /\ Fun `'(F |` C)))
14	10, 12, 13	3imtr4 219	1 |- ((F:A-1-1->B /\ C (_ A) -> (F |` C):C-1-1-onto->(F"C))

Syntax hints:   -> wi 3   /\ wa 223   (_ wss 2047  `'ccnv 3169  dom cdm 3170   |` cres 3172  "cima 3173  Fun wfun 3176  -->wf 3178  -1-1->wf1 3179  -onto->wfo 3180  -1-1-onto->wf1o 3181

This theorem is referenced by:  f1imacnv 3705  f1imaen 4422  phplem4 4511  php3 4515  php3OLD 4516  ssfi 4537  ssfiOLD 4538  unifiOLD 4557  fiint 4559  fiintOLD 4560  unbenlem 7504  adjbd1o 10018

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197

Copyright terms: Public domain