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Theorem f1ocnvfv3 6199
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 6101 . . 3  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( `' F `  C )  e.  A
)
2 f1ocnvfvb 6098 . . . . . 6  |-  ( ( F : A -1-1-onto-> B  /\  x  e.  A  /\  C  e.  B )  ->  ( ( F `  x )  =  C  <-> 
( `' F `  C )  =  x ) )
323expa 1188 . . . . 5  |-  ( ( ( F : A -1-1-onto-> B  /\  x  e.  A
)  /\  C  e.  B )  ->  (
( F `  x
)  =  C  <->  ( `' F `  C )  =  x ) )
43an32s 802 . . . 4  |-  ( ( ( F : A -1-1-onto-> B  /\  C  e.  B
)  /\  x  e.  A )  ->  (
( F `  x
)  =  C  <->  ( `' F `  C )  =  x ) )
5 eqcom 2463 . . . 4  |-  ( x  =  ( `' F `  C )  <->  ( `' F `  C )  =  x )
64, 5syl6bbr 263 . . 3  |-  ( ( ( F : A -1-1-onto-> B  /\  C  e.  B
)  /\  x  e.  A )  ->  (
( F `  x
)  =  C  <->  x  =  ( `' F `  C ) ) )
71, 6riota5 6190 . 2  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( iota_ x  e.  A  ( F `  x )  =  C )  =  ( `' F `  C ) )
87eqcomd 2462 1  |-  ( ( F : A -1-1-onto-> B  /\  C  e.  B )  ->  ( `' F `  C )  =  (
iota_ x  e.  A  ( F `  x )  =  C ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   `'ccnv 4950   -1-1-onto->wf1o 5528   ` cfv 5529   iota_crio 6163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164
This theorem is referenced by: (None)
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