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Theorem foelrn 5917
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 5913 . . 3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
21simprbi 452 . 2  |-  ( F : A -onto-> B  ->  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) )
3 eqeq1 2448 . . . 4  |-  ( y  =  C  ->  (
y  =  ( F `
 x )  <->  C  =  ( F `  x ) ) )
43rexbidv 2732 . . 3  |-  ( y  =  C  ->  ( E. x  e.  A  y  =  ( F `  x )  <->  E. x  e.  A  C  =  ( F `  x ) ) )
54rspccva 3057 . 2  |-  ( ( A. y  e.  B  E. x  e.  A  y  =  ( F `  x )  /\  C  e.  B )  ->  E. x  e.  A  C  =  ( F `  x ) )
62, 5sylan 459 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 360    = wceq 1653    e. wcel 1727   A.wral 2711   E.wrex 2712   -->wf 5479   -onto->wfo 5481   ` cfv 5483
This theorem is referenced by:  foco2  5918  fofinf1o  7416  fodomacn  7968  iunfictbso  8026  cff1  8169  cofsmo  8180  axcclem  8368  konigthlem  8474  tskuni  8689  fulli  14141  efgredlemc  15408  efgrelexlemb  15413  efgredeu  15415  ghmcyg  15536  znfld  16872  znrrg  16877  cygznlem3  16881  ovoliunnul  19434  lgsdchr  21163  ghgrplem1  21985  iunrdx  24045  crngohomfo  26654
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pr 4432
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fo 5489  df-fv 5491
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