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Theorem f1dmex 6131
Description: If the codomain of a one-to-one function exists, so does its domain. This can be thought of as a form of the Axiom of Replacement. (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 f1rn 5434 . . . . 5  |-  ( F : A -1-1-> B  ->  ran  F  C_  B )
2 ssexg 4154 . . . . 5  |-  ( ( ran  F  C_  B  /\  B  e.  C
)  ->  ran  F  e. 
_V )
31, 2sylan 283 . . . 4  |-  ( ( F : A -1-1-> B  /\  B  e.  C
)  ->  ran  F  e. 
_V )
43ex 115 . . 3  |-  ( F : A -1-1-> B  -> 
( B  e.  C  ->  ran  F  e.  _V ) )
5 f1cnv 5497 . . . . 5  |-  ( F : A -1-1-> B  ->  `' F : ran  F -1-1-onto-> A
)
6 f1ofo 5480 . . . . 5  |-  ( `' F : ran  F -1-1-onto-> A  ->  `' F : ran  F -onto-> A )
75, 6syl 14 . . . 4  |-  ( F : A -1-1-> B  ->  `' F : ran  F -onto-> A )
8 focdmex 6130 . . . 4  |-  ( ran 
F  e.  _V  ->  ( `' F : ran  F -onto-> A  ->  A  e.  _V ) )
97, 8syl5com 29 . . 3  |-  ( F : A -1-1-> B  -> 
( ran  F  e.  _V  ->  A  e.  _V ) )
104, 9syld 45 . 2  |-  ( F : A -1-1-> B  -> 
( B  e.  C  ->  A  e.  _V )
)
1110imp 124 1  |-  ( ( F : A -1-1-> B  /\  B  e.  C
)  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    e. wcel 2158   _Vcvv 2749    C_ wss 3141   `'ccnv 4637   ran crn 4639   -1-1->wf1 5225   -onto->wfo 5226   -1-1-onto->wf1o 5227
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236
This theorem is referenced by:  f1domg  6772
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