Theorem f1ores 3698

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 3676	. . . . 5 |- ((Fun F /\ C (_ dom F) -> (F |` C):C-onto->(F"C))
2		ffun 3625	. . . . . 6 |- (F:A-->B -> Fun F)
3	2	adantr 389	. . . . 5 |- ((F:A-->B /\ C (_ A) -> Fun F)
4		fdm 3627	. . . . . . 7 |- (F:A-->B -> dom F = A)
5	4	sseq2d 2086	. . . . . 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 3563	. . . 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 3191	. . 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 3689	. 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 2044  `'ccnv 3165  dom cdm 3166   |` cres 3168  "cima 3169  Fun wfun 3172  -->wf 3174  -1-1->wf1 3175  -onto->wfo 3176  -1-1-onto->wf1o 3177

This theorem is referenced by:  f1imacnv 3700  f1imaen 4412  phplem4 4500  php3 4504  ssfi 4524  unifi 4541  fiint 4543  unbenlem 7464  adjbd1o 9974

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-f1 3191  df-fo 3192  df-f1o 3193

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