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Theorem f1ocnvfv3 6340
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 5796 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( `' F `  C )  e.  A
)
2 f1ocnvfvb 5795 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  x  e.  A  /\  C  e.  B )  ->  ( ( F `  x )  =  C  <-> 
( `' F `  C )  =  x ) )
323expa 1151 . . . . 5  |-  ( ( ( F : A -1-1-onto-> B  /\  x  e.  A
)  /\  C  e.  B )  ->  (
( F `  x
)  =  C  <->  ( `' F `  C )  =  x ) )
43an32s 779 . . . 4  |-  ( ( ( F : A -1-1-onto-> B  /\  C  e.  B
)  /\  x  e.  A )  ->  (
( F `  x
)  =  C  <->  ( `' F `  C )  =  x ) )
5 eqcom 2285 . . . 4  |-  ( x  =  ( `' F `  C )  <->  ( `' F `  C )  =  x )
64, 5syl6bbr 254 . . 3  |-  ( ( ( F : A -1-1-onto-> B  /\  C  e.  B
)  /\  x  e.  A )  ->  (
( F `  x
)  =  C  <->  x  =  ( `' F `  C ) ) )
71, 6riota5 6330 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( iota_ x  e.  A
( F `  x
)  =  C )  =  ( `' F `  C ) )
87eqcomd 2288 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    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   `'ccnv 4688   -1-1-onto->wf1o 5254   ` cfv 5255   iota_crio 6297
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 6304
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