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Theorem f1ocnvfv3 5877
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 5795 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( `' F `  C )  e.  A
)
2 f1ocnvfvb 5794 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  x  e.  A  /\  C  e.  B )  ->  ( ( F `  x )  =  C  <-> 
( `' F `  C )  =  x ) )
323expa 1204 . . . . 5  |-  ( ( ( F : A -1-1-onto-> B  /\  x  e.  A
)  /\  C  e.  B )  ->  (
( F `  x
)  =  C  <->  ( `' F `  C )  =  x ) )
43an32s 568 . . . 4  |-  ( ( ( F : A -1-1-onto-> B  /\  C  e.  B
)  /\  x  e.  A )  ->  (
( F `  x
)  =  C  <->  ( `' F `  C )  =  x ) )
5 eqcom 2189 . . . 4  |-  ( x  =  ( `' F `  C )  <->  ( `' F `  C )  =  x )
64, 5bitr4di 198 . . 3  |-  ( ( ( F : A -1-1-onto-> B  /\  C  e.  B
)  /\  x  e.  A )  ->  (
( F `  x
)  =  C  <->  x  =  ( `' F `  C ) ) )
71, 6riota5 5869 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( iota_ x  e.  A  ( F `  x )  =  C )  =  ( `' F `  C ) )
87eqcomd 2193 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 104    <-> wb 105    = wceq 1363    e. wcel 2158   `'ccnv 4637   -1-1-onto->wf1o 5227   ` cfv 5228   iota_crio 5843
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470  df-rex 2471  df-reu 2472  df-v 2751  df-sbc 2975  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-riota 5844
This theorem is referenced by: (None)
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