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Theorem f1dmex 5930
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 4280. (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 5598 . . . . . 6  |-  ( F : A -1-1-> B  ->  F : A --> B )
2 frn 5556 . . . . . 6  |-  ( F : A --> B  ->  ran  F  C_  B )
31, 2syl 16 . . . . 5  |-  ( F : A -1-1-> B  ->  ran  F  C_  B )
4 ssexg 4309 . . . . 5  |-  ( ( ran  F  C_  B  /\  B  e.  C
)  ->  ran  F  e. 
_V )
53, 4sylan 458 . . . 4  |-  ( ( F : A -1-1-> B  /\  B  e.  C
)  ->  ran  F  e. 
_V )
65ex 424 . . 3  |-  ( F : A -1-1-> B  -> 
( B  e.  C  ->  ran  F  e.  _V ) )
7 f1cnv 5658 . . . . 5  |-  ( F : A -1-1-> B  ->  `' F : ran  F -1-1-onto-> A
)
8 f1ofo 5640 . . . . 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 5929 . . . 4  |-  ( ran 
F  e.  _V  ->  ( `' F : ran  F -onto-> A  ->  A  e.  _V ) )
119, 10syl5com 28 . . 3  |-  ( F : A -1-1-> B  -> 
( ran  F  e.  _V  ->  A  e.  _V ) )
126, 11syld 42 . 2  |-  ( F : A -1-1-> B  -> 
( B  e.  C  ->  A  e.  _V )
)
1312imp 419 1  |-  ( ( F : A -1-1-> B  /\  B  e.  C
)  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721   _Vcvv 2916    C_ wss 3280   `'ccnv 4836   ran crn 4838   -->wf 5409   -1-1->wf1 5410   -onto->wfo 5411   -1-1-onto->wf1o 5412
This theorem is referenced by:  abianfp  6675  f1domg  7086  ordtypelem10  7452  oiexg  7460  inf3lem7  7545  pwfseqlem4  8493  pwfseqlem5  8494  grothomex  8660  gsumzf1o  15474  dprdf1o  15545  tsmsf1o  18127  diophrw  26707  f1lindf  27160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421
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