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Theorem f1dmex 5982
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 5299 . . . . 5  |-  ( F : A -1-1-> B  ->  ran  F  C_  B )
2 ssexg 4037 . . . . 5  |-  ( ( ran  F  C_  B  /\  B  e.  C
)  ->  ran  F  e. 
_V )
31, 2sylan 281 . . . 4  |-  ( ( F : A -1-1-> B  /\  B  e.  C
)  ->  ran  F  e. 
_V )
43ex 114 . . 3  |-  ( F : A -1-1-> B  -> 
( B  e.  C  ->  ran  F  e.  _V ) )
5 f1cnv 5359 . . . . 5  |-  ( F : A -1-1-> B  ->  `' F : ran  F -1-1-onto-> A
)
6 f1ofo 5342 . . . . 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 fornex 5981 . . . 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 123 1  |-  ( ( F : A -1-1-> B  /\  B  e.  C
)  ->  A  e.  _V )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1465   _Vcvv 2660    C_ wss 3041   `'ccnv 4508   ran crn 4510   -1-1->wf1 5090   -onto->wfo 5091   -1-1-onto->wf1o 5092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101
This theorem is referenced by:  f1domg  6620
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