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Theorem fornex 5933
Description: If the domain of an onto function exists, so does its codomain. (Contributed by NM, 23-Jul-2004.)
Assertion
Ref Expression
fornex  |-  ( A  e.  C  ->  ( F : A -onto-> B  ->  B  e.  _V )
)

Proof of Theorem fornex
StepHypRef Expression
1 fofun 5617 . . . 4  |-  ( F : A -onto-> B  ->  Fun  F )
2 funrnex 5930 . . . 4  |-  ( dom 
F  e.  C  -> 
( Fun  F  ->  ran 
F  e.  _V )
)
31, 2syl5com 28 . . 3  |-  ( F : A -onto-> B  -> 
( dom  F  e.  C  ->  ran  F  e.  _V ) )
4 fof 5616 . . . . 5  |-  ( F : A -onto-> B  ->  F : A --> B )
5 fdm 5558 . . . . 5  |-  ( F : A --> B  ->  dom  F  =  A )
64, 5syl 16 . . . 4  |-  ( F : A -onto-> B  ->  dom  F  =  A )
76eleq1d 2474 . . 3  |-  ( F : A -onto-> B  -> 
( dom  F  e.  C 
<->  A  e.  C ) )
8 forn 5619 . . . 4  |-  ( F : A -onto-> B  ->  ran  F  =  B )
98eleq1d 2474 . . 3  |-  ( F : A -onto-> B  -> 
( ran  F  e.  _V 
<->  B  e.  _V )
)
103, 7, 93imtr3d 259 . 2  |-  ( F : A -onto-> B  -> 
( A  e.  C  ->  B  e.  _V )
)
1110com12 29 1  |-  ( A  e.  C  ->  ( F : A -onto-> B  ->  B  e.  _V )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   _Vcvv 2920   dom cdm 4841   ran crn 4842   Fun wfun 5411   -->wf 5413   -onto->wfo 5415
This theorem is referenced by:  f1dmex  5934  f1oeng  7089  fodomnum  7898  ttukeylem1  8349  fodomb  8364  cnexALT  10568  imasbas  13697  imasds  13698  elqtop  17686  qtoprest  17706  indishmph  17787  imasf1oxmet  18362  ghgrp  21913  noprc  25553
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 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pr 4367  ax-un 4664
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 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-iun 4059  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425
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