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Theorem foelcdmi 5570
Description: A member of a surjective function's codomain is a value of the function. (Contributed by Thierry Arnoux, 23-Jan-2020.)
Assertion
Ref Expression
foelcdmi  |-  ( ( F : A -onto-> B  /\  Y  e.  B
)  ->  E. x  e.  A  ( F `  x )  =  Y )
Distinct variable groups:    x, A    x, B    x, F    x, Y

Proof of Theorem foelcdmi
StepHypRef Expression
1 forn 5443 . . . 4  |-  ( F : A -onto-> B  ->  ran  F  =  B )
21eleq2d 2247 . . 3  |-  ( F : A -onto-> B  -> 
( Y  e.  ran  F  <-> 
Y  e.  B ) )
3 fofn 5442 . . . 4  |-  ( F : A -onto-> B  ->  F  Fn  A )
4 fvelrnb 5565 . . . 4  |-  ( F  Fn  A  ->  ( Y  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  Y ) )
53, 4syl 14 . . 3  |-  ( F : A -onto-> B  -> 
( Y  e.  ran  F  <->  E. x  e.  A  ( F `  x )  =  Y ) )
62, 5bitr3d 190 . 2  |-  ( F : A -onto-> B  -> 
( Y  e.  B  <->  E. x  e.  A  ( F `  x )  =  Y ) )
76biimpa 296 1  |-  ( ( F : A -onto-> B  /\  Y  e.  B
)  ->  E. x  e.  A  ( F `  x )  =  Y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353    e. wcel 2148   E.wrex 2456   ran crn 4629    Fn wfn 5213   -onto->wfo 5216   ` cfv 5218
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-fo 5224  df-fv 5226
This theorem is referenced by:  mhmid  12984  mhmmnd  12985  ghmgrp  12987
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