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Theorem fornex 5870
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 5558 . . . 4  |-  ( F : A -onto-> B  ->  Fun  F )
2 funrnex 5867 . . . 4  |-  ( dom 
F  e.  C  -> 
( Fun  F  ->  ran 
F  e.  _V )
)
31, 2syl5com 26 . . 3  |-  ( F : A -onto-> B  -> 
( dom  F  e.  C  ->  ran  F  e.  _V ) )
4 fof 5557 . . . . 5  |-  ( F : A -onto-> B  ->  F : A --> B )
5 fdm 5499 . . . . 5  |-  ( F : A --> B  ->  dom  F  =  A )
64, 5syl 15 . . . 4  |-  ( F : A -onto-> B  ->  dom  F  =  A )
76eleq1d 2432 . . 3  |-  ( F : A -onto-> B  -> 
( dom  F  e.  C 
<->  A  e.  C ) )
8 forn 5560 . . . 4  |-  ( F : A -onto-> B  ->  ran  F  =  B )
98eleq1d 2432 . . 3  |-  ( F : A -onto-> B  -> 
( ran  F  e.  _V 
<->  B  e.  _V )
)
103, 7, 93imtr3d 258 . 2  |-  ( F : A -onto-> B  -> 
( A  e.  C  ->  B  e.  _V )
)
1110com12 27 1  |-  ( A  e.  C  ->  ( F : A -onto-> B  ->  B  e.  _V )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1647    e. wcel 1715   _Vcvv 2873   dom cdm 4792   ran crn 4793   Fun wfun 5352   -->wf 5354   -onto->wfo 5356
This theorem is referenced by:  f1dmex  5871  f1oeng  7023  fodomnum  7831  ttukeylem1  8283  fodomb  8298  cnexALT  10501  imasbas  13625  imasds  13626  elqtop  17605  qtoprest  17625  indishmph  17706  imasf1oxmet  18152  ghgrp  21467  noprc  25161
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366
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