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Theorem f1dmex 5974
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 4323. (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 5642 . . . . . 6  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 frn 5600 . . . . . 6  |-  ( F : A --> B  ->  ran  F  C_  B )
31, 2syl 16 . . . . 5  |-  ( F : A -1-1-> B  ->  ran  F  C_  B )
4 ssexg 4352 . . . . 5  |-  ( ( ran  F  C_  B  /\  B  e.  C
)  ->  ran  F  e. 
_V )
53, 4sylan 459 . . . 4  |-  ( ( F : A -1-1-> B  /\  B  e.  C
)  ->  ran  F  e. 
_V )
65ex 425 . . 3  |-  ( F : A -1-1-> B  -> 
( B  e.  C  ->  ran  F  e.  _V ) )
7 f1cnv 5702 . . . . 5  |-  ( F : A -1-1-> B  ->  `' F : ran  F -1-1-onto-> A
)
8 f1ofo 5684 . . . . 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 5973 . . . 4  |-  ( ran 
F  e.  _V  ->  ( `' F : ran  F -onto-> A  ->  A  e.  _V ) )
119, 10syl5com 29 . . 3  |-  ( F : A -1-1-> B  -> 
( ran  F  e.  _V  ->  A  e.  _V ) )
126, 11syld 43 . 2  |-  ( F : A -1-1-> B  -> 
( B  e.  C  ->  A  e.  _V )
)
1312imp 420 1  |-  ( ( F : A -1-1-> B  /\  B  e.  C
)  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1726   _Vcvv 2958    C_ wss 3322   `'ccnv 4880   ran crn 4882   -->wf 5453   -1-1->wf1 5454   -onto->wfo 5455   -1-1-onto->wf1o 5456
This theorem is referenced by:  abianfp  6719  f1domg  7130  ordtypelem10  7499  oiexg  7507  inf3lem7  7592  pwfseqlem4  8542  pwfseqlem5  8543  grothomex  8709  gsumzf1o  15524  dprdf1o  15595  tsmsf1o  18179  diophrw  26831  f1lindf  27283
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465
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