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Theorem f1dmex 6746
Description: If the codomain of a one-to-one function exists, so does its domain. This theorem is equivalent to the Axiom of Replacement ax-rep 4553. (Contributed by NM, 4-Sep-2004.)
Assertion
Ref Expression
f1dmex  |-  ( ( F : A -1-1-> B  /\  B  e.  C
)  ->  A  e.  _V )

Proof of Theorem f1dmex
StepHypRef Expression
1 f1f 5774 . . . . . 6  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 frn 5730 . . . . . 6  |-  ( F : A --> B  ->  ran  F  C_  B )
31, 2syl 16 . . . . 5  |-  ( F : A -1-1-> B  ->  ran  F  C_  B )
4 ssexg 4588 . . . . 5  |-  ( ( ran  F  C_  B  /\  B  e.  C
)  ->  ran  F  e. 
_V )
53, 4sylan 471 . . . 4  |-  ( ( F : A -1-1-> B  /\  B  e.  C
)  ->  ran  F  e. 
_V )
65ex 434 . . 3  |-  ( F : A -1-1-> B  -> 
( B  e.  C  ->  ran  F  e.  _V ) )
7 f1cnv 5832 . . . . 5  |-  ( F : A -1-1-> B  ->  `' F : ran  F -1-1-onto-> A
)
8 f1ofo 5816 . . . . 5  |-  ( `' F : ran  F -1-1-onto-> A  ->  `' F : ran  F -onto-> A )
97, 8syl 16 . . . 4  |-  ( F : A -1-1-> B  ->  `' F : ran  F -onto-> A )
10 fornex 6745 . . . 4  |-  ( ran 
F  e.  _V  ->  ( `' F : ran  F -onto-> A  ->  A  e.  _V ) )
119, 10syl5com 30 . . 3  |-  ( F : A -1-1-> B  -> 
( ran  F  e.  _V  ->  A  e.  _V ) )
126, 11syld 44 . 2  |-  ( F : A -1-1-> B  -> 
( B  e.  C  ->  A  e.  _V )
)
1312imp 429 1  |-  ( ( F : A -1-1-> B  /\  B  e.  C
)  ->  A  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1762   _Vcvv 3108    C_ wss 3471   `'ccnv 4993   ran crn 4995   -->wf 5577   -1-1->wf1 5578   -onto->wfo 5579   -1-1-onto->wf1o 5580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-rep 4553  ax-sep 4563  ax-nul 4571  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-reu 2816  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589
This theorem is referenced by:  f1ovv  6747  f1domg  7527  ordtypelem10  7943  oiexg  7951  inf3lem7  8042  pwfseqlem4  9031  pwfseqlem5  9032  grothomex  9198  gsumzf1o  16703  gsumzf1oOLD  16706  dprdf1o  16864  f1lindf  18619  tsmsf1o  20377  diophrw  30285
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