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Theorem foelrn 5345
Description: Property of a surjective function. (Contributed by Jeff Madsen, 4-Jan-2011.)
Assertion
Ref Expression
foelrn  |-  ( ( F : A -onto-> B  /\  C  e.  B
)  ->  E. x  e.  A  C  =  ( F `  x ) )
Distinct variable groups:    x, F    x, A    x, B    x, C

Proof of Theorem foelrn
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 dffo3 5342 . . 3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
21simprbi 264 . 2  |-  ( F : A -onto-> B  ->  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) )
3 eqeq1 2062 . . . 4  |-  ( y  =  C  ->  (
y  =  ( F `
 x )  <->  C  =  ( F `  x ) ) )
43rexbidv 2344 . . 3  |-  ( y  =  C  ->  ( E. x  e.  A  y  =  ( F `  x )  <->  E. x  e.  A  C  =  ( F `  x ) ) )
54rspccva 2672 . 2  |-  ( ( A. y  e.  B  E. x  e.  A  y  =  ( F `  x )  /\  C  e.  B )  ->  E. x  e.  A  C  =  ( F `  x ) )
62, 5sylan 271 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 101    = wceq 1259    e. wcel 1409   A.wral 2323   E.wrex 2324   -->wf 4926   -onto->wfo 4928   ` cfv 4930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-fo 4936  df-fv 4938
This theorem is referenced by:  foco2  5346
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