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Theorem foelrn 5952
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 5948 . . 3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
21simprbi 462 . 2  |-  ( F : A -onto-> B  ->  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) )
3 eqeq1 2386 . . . 4  |-  ( y  =  C  ->  (
y  =  ( F `
 x )  <->  C  =  ( F `  x ) ) )
43rexbidv 2893 . . 3  |-  ( y  =  C  ->  ( E. x  e.  A  y  =  ( F `  x )  <->  E. x  e.  A  C  =  ( F `  x ) ) )
54rspccva 3134 . 2  |-  ( ( A. y  e.  B  E. x  e.  A  y  =  ( F `  x )  /\  C  e.  B )  ->  E. x  e.  A  C  =  ( F `  x ) )
62, 5sylan 469 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 367    = wceq 1399    e. wcel 1826   A.wral 2732   E.wrex 2733   -->wf 5492   -onto->wfo 5494   ` cfv 5496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fo 5502  df-fv 5504
This theorem is referenced by:  foco2  5953  fofinf1o  7716  fodomacn  8350  iunfictbso  8408  cff1  8551  cofsmo  8562  axcclem  8750  konigthlem  8856  tskuni  9072  fulli  15319  efgredlemc  16880  efgrelexlemb  16885  efgredeu  16887  ghmcyg  17015  znfld  18690  znrrg  18695  cygznlem3  18699  ovoliunnul  22003  lgsdchr  23740  ghgrplem1OLD  25485  foresf1o  27521  iunrdx  27560  crngohomfo  30569  fourierdlem20  32075  fourierdlem52  32107  fourierdlem63  32118  fourierdlem64  32119  fourierdlem65  32120
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