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Theorem foelrn 5620
Description: Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.) (Proof shortened by BJ, 6-Jul-2022.)
Assertion
Ref Expression
foelrn  |-  ( ( F : A -onto-> B  /\  C  e.  B
)  ->  E. x  e.  A  C  =  ( F `  x ) )
Distinct variable groups:    x, A    x, C    x, F
Allowed substitution hint:    B( x)

Proof of Theorem foelrn
StepHypRef Expression
1 foima2 5619 . 2  |-  ( F : A -onto-> B  -> 
( C  e.  B  <->  E. x  e.  A  C  =  ( F `  x ) ) )
21biimpa 292 1  |-  ( ( F : A -onto-> B  /\  C  e.  B
)  ->  E. x  e.  A  C  =  ( F `  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463   E.wrex 2392   -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:  foco2  5621  ctmlemr  6959  ctm  6960  ctssdclemn0  6961  ctssdccl  6962  ctssdc  6964  enumctlemm  6965  fodju0  6985  exmidfodomrlemr  7022  exmidfodomrlemrALT  7023  ennnfonelemrn  11827  ctinf  11838  ctiunctlemfo  11847  subctctexmid  13007
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