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Theorem foima2 5530
Description: Given an onto function, an element is in its codomain if and only if it is the image of an element of its domain (see foima 5238). (Contributed by BJ, 6-Jul-2022.)
Assertion
Ref Expression
foima2  |-  ( F : A -onto-> B  -> 
( Y  e.  B  <->  E. x  e.  A  Y  =  ( F `  x ) ) )
Distinct variable groups:    x, A    x, Y    x, F
Allowed substitution hint:    B( x)

Proof of Theorem foima2
StepHypRef Expression
1 foima 5238 . . . 4  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )
21eqcomd 2093 . . 3  |-  ( F : A -onto-> B  ->  B  =  ( F " A ) )
32eleq2d 2157 . 2  |-  ( F : A -onto-> B  -> 
( Y  e.  B  <->  Y  e.  ( F " A ) ) )
4 fofn 5235 . . 3  |-  ( F : A -onto-> B  ->  F  Fn  A )
5 ssid 3044 . . 3  |-  A  C_  A
6 fvelimab 5360 . . . 4  |-  ( ( F  Fn  A  /\  A  C_  A )  -> 
( Y  e.  ( F " A )  <->  E. x  e.  A  ( F `  x )  =  Y ) )
7 eqcom 2090 . . . . 5  |-  ( ( F `  x )  =  Y  <->  Y  =  ( F `  x ) )
87rexbii 2385 . . . 4  |-  ( E. x  e.  A  ( F `  x )  =  Y  <->  E. x  e.  A  Y  =  ( F `  x ) )
96, 8syl6bb 194 . . 3  |-  ( ( F  Fn  A  /\  A  C_  A )  -> 
( Y  e.  ( F " A )  <->  E. x  e.  A  Y  =  ( F `  x ) ) )
104, 5, 9sylancl 404 . 2  |-  ( F : A -onto-> B  -> 
( Y  e.  ( F " A )  <->  E. x  e.  A  Y  =  ( F `  x ) ) )
113, 10bitrd 186 1  |-  ( F : A -onto-> B  -> 
( Y  e.  B  <->  E. x  e.  A  Y  =  ( F `  x ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   E.wrex 2360    C_ wss 2999   "cima 4441    Fn wfn 5010   -onto->wfo 5013   ` cfv 5015
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fo 5021  df-fv 5023
This theorem is referenced by:  foelrn  5531
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