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Theorem f1ocnvfv3 5641
Description: Value of the converse of a one-to-one onto function. (Contributed by NM, 26-May-2006.) (Proof shortened by Mario Carneiro, 24-Dec-2016.)
Assertion
Ref Expression
f1ocnvfv3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( `' F `  C )  =  (
iota_ x  e.  A  ( F `  x )  =  C ) )
Distinct variable groups:    x, A    x, B    x, C    x, F

Proof of Theorem f1ocnvfv3
StepHypRef Expression
1 f1ocnvdm 5560 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( `' F `  C )  e.  A
)
2 f1ocnvfvb 5559 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  x  e.  A  /\  C  e.  B )  ->  ( ( F `  x )  =  C  <-> 
( `' F `  C )  =  x ) )
323expa 1143 . . . . 5  |-  ( ( ( F : A -1-1-onto-> B  /\  x  e.  A
)  /\  C  e.  B )  ->  (
( F `  x
)  =  C  <->  ( `' F `  C )  =  x ) )
43an32s 535 . . . 4  |-  ( ( ( F : A -1-1-onto-> B  /\  C  e.  B
)  /\  x  e.  A )  ->  (
( F `  x
)  =  C  <->  ( `' F `  C )  =  x ) )
5 eqcom 2090 . . . 4  |-  ( x  =  ( `' F `  C )  <->  ( `' F `  C )  =  x )
64, 5syl6bbr 196 . . 3  |-  ( ( ( F : A -1-1-onto-> B  /\  C  e.  B
)  /\  x  e.  A )  ->  (
( F `  x
)  =  C  <->  x  =  ( `' F `  C ) ) )
71, 6riota5 5633 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( iota_ x  e.  A  ( F `  x )  =  C )  =  ( `' F `  C ) )
87eqcomd 2093 1  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( `' F `  C )  =  (
iota_ x  e.  A  ( F `  x )  =  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103    = wceq 1289    e. wcel 1438   `'ccnv 4437   -1-1-onto->wf1o 5014   ` cfv 5015   iota_crio 5607
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-reu 2366  df-v 2621  df-sbc 2841  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-f1 5020  df-fo 5021  df-f1o 5022  df-fv 5023  df-riota 5608
This theorem is referenced by: (None)
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