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Theorem foima2 5619
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 5318). (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 5318 . . . 4  |-  ( F : A -onto-> B  -> 
( F " A
)  =  B )
21eqcomd 2121 . . 3  |-  ( F : A -onto-> B  ->  B  =  ( F " A ) )
32eleq2d 2185 . 2  |-  ( F : A -onto-> B  -> 
( Y  e.  B  <->  Y  e.  ( F " A ) ) )
4 fofn 5315 . . 3  |-  ( F : A -onto-> B  ->  F  Fn  A )
5 ssid 3085 . . 3  |-  A  C_  A
6 fvelimab 5443 . . . 4  |-  ( ( F  Fn  A  /\  A  C_  A )  -> 
( Y  e.  ( F " A )  <->  E. x  e.  A  ( F `  x )  =  Y ) )
7 eqcom 2117 . . . . 5  |-  ( ( F `  x )  =  Y  <->  Y  =  ( F `  x ) )
87rexbii 2417 . . . 4  |-  ( E. x  e.  A  ( F `  x )  =  Y  <->  E. x  e.  A  Y  =  ( F `  x ) )
96, 8syl6bb 195 . . 3  |-  ( ( F  Fn  A  /\  A  C_  A )  -> 
( Y  e.  ( F " A )  <->  E. x  e.  A  Y  =  ( F `  x ) ) )
104, 5, 9sylancl 407 . 2  |-  ( F : A -onto-> B  -> 
( Y  e.  ( F " A )  <->  E. x  e.  A  Y  =  ( F `  x ) ) )
113, 10bitrd 187 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 103    <-> wb 104    = wceq 1314    e. wcel 1463   E.wrex 2392    C_ wss 3039   "cima 4510    Fn wfn 5086   -onto->wfo 5089   ` cfv 5091
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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-sbc 2881  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fo 5097  df-fv 5099
This theorem is referenced by:  foelrn  5620
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