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Theorem foelrn 5862
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 5858 . . 3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
21simprbi 464 . 2  |-  ( F : A -onto-> B  ->  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) )
3 eqeq1 2449 . . . 4  |-  ( y  =  C  ->  (
y  =  ( F `
 x )  <->  C  =  ( F `  x ) ) )
43rexbidv 2736 . . 3  |-  ( y  =  C  ->  ( E. x  e.  A  y  =  ( F `  x )  <->  E. x  e.  A  C  =  ( F `  x ) ) )
54rspccva 3072 . 2  |-  ( ( A. y  e.  B  E. x  e.  A  y  =  ( F `  x )  /\  C  e.  B )  ->  E. x  e.  A  C  =  ( F `  x ) )
62, 5sylan 471 1  |-  ( ( F : A -onto-> B  /\  C  e.  B
)  ->  E. x  e.  A  C  =  ( F `  x ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715   E.wrex 2716   -->wf 5414   -onto->wfo 5416   ` cfv 5418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-fo 5424  df-fv 5426
This theorem is referenced by:  foco2  5863  fofinf1o  7592  fodomacn  8226  iunfictbso  8284  cff1  8427  cofsmo  8438  axcclem  8626  konigthlem  8732  tskuni  8950  fulli  14823  symgmov2  15898  efgredlemc  16242  efgrelexlemb  16247  efgredeu  16249  ghmcyg  16372  znfld  17993  znrrg  17998  cygznlem3  18002  ovoliunnul  20990  lgsdchr  22687  ghgrplem1  23853  iunrdx  25914  crngohomfo  28806
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