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Theorem f1ocnvfv3 5731
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 5650 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( `' F `  C )  e.  A
)
2 f1ocnvfvb 5649 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  x  e.  A  /\  C  e.  B )  ->  ( ( F `  x )  =  C  <-> 
( `' F `  C )  =  x ) )
323expa 1166 . . . . 5  |-  ( ( ( F : A -1-1-onto-> B  /\  x  e.  A
)  /\  C  e.  B )  ->  (
( F `  x
)  =  C  <->  ( `' F `  C )  =  x ) )
43an32s 542 . . . 4  |-  ( ( ( F : A -1-1-onto-> B  /\  C  e.  B
)  /\  x  e.  A )  ->  (
( F `  x
)  =  C  <->  ( `' F `  C )  =  x ) )
5 eqcom 2119 . . . 4  |-  ( x  =  ( `' F `  C )  <->  ( `' F `  C )  =  x )
64, 5syl6bbr 197 . . 3  |-  ( ( ( F : A -1-1-onto-> B  /\  C  e.  B
)  /\  x  e.  A )  ->  (
( F `  x
)  =  C  <->  x  =  ( `' F `  C ) ) )
71, 6riota5 5723 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( iota_ x  e.  A  ( F `  x )  =  C )  =  ( `' F `  C ) )
87eqcomd 2123 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 103    <-> wb 104    = wceq 1316    e. wcel 1465   `'ccnv 4508   -1-1-onto->wf1o 5092   ` cfv 5093   iota_crio 5697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-reu 2400  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-riota 5698
This theorem is referenced by: (None)
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