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Theorem foelrnf 37461
Description: Property of a surjective function. As foelrn 6041 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypothesis
Ref Expression
foelrnf.1  |-  F/_ x F
Assertion
Ref Expression
foelrnf  |-  ( ( F : A -onto-> B  /\  C  e.  B
)  ->  E. x  e.  A  C  =  ( F `  x ) )
Distinct variable groups:    x, A    x, B    x, C
Allowed substitution hint:    F( x)

Proof of Theorem foelrnf
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 foelrnf.1 . . . 4  |-  F/_ x F
21dffo3f 37450 . . 3  |-  ( F : A -onto-> B  <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
32simprbi 466 . 2  |-  ( F : A -onto-> B  ->  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) )
4 eqeq1 2455 . . . 4  |-  ( y  =  C  ->  (
y  =  ( F `
 x )  <->  C  =  ( F `  x ) ) )
54rexbidv 2901 . . 3  |-  ( y  =  C  ->  ( E. x  e.  A  y  =  ( F `  x )  <->  E. x  e.  A  C  =  ( F `  x ) ) )
65rspccva 3149 . 2  |-  ( ( A. y  e.  B  E. x  e.  A  y  =  ( F `  x )  /\  C  e.  B )  ->  E. x  e.  A  C  =  ( F `  x ) )
73, 6sylan 474 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 371    = wceq 1444    e. wcel 1887   F/_wnfc 2579   A.wral 2737   E.wrex 2738   -->wf 5578   -onto->wfo 5580   ` cfv 5582
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pr 4639
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-fo 5588  df-fv 5590
This theorem is referenced by:  fompt  37467
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