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Theorem fv1stcnv 30470
Description: The value of the converse of  1st restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)
Assertion
Ref Expression
fv1stcnv  |-  ( ( X  e.  A  /\  Y  e.  V )  ->  ( `' ( 1st  |`  ( A  X.  { Y } ) ) `  X )  =  <. X ,  Y >. )

Proof of Theorem fv1stcnv
StepHypRef Expression
1 snidg 4005 . . . . 5  |-  ( Y  e.  V  ->  Y  e.  { Y } )
21anim2i 577 . . . 4  |-  ( ( X  e.  A  /\  Y  e.  V )  ->  ( X  e.  A  /\  Y  e.  { Y } ) )
3 eqid 2461 . . . 4  |-  X  =  X
42, 3jctil 544 . . 3  |-  ( ( X  e.  A  /\  Y  e.  V )  ->  ( X  =  X  /\  ( X  e.  A  /\  Y  e. 
{ Y } ) ) )
5 opex 4677 . . . . . . 7  |-  <. X ,  Y >.  e.  _V
6 brcnvg 5033 . . . . . . 7  |-  ( ( X  e.  A  /\  <. X ,  Y >.  e. 
_V )  ->  ( X `' ( 1st  |`  ( A  X.  { Y }
) ) <. X ,  Y >. 
<-> 
<. X ,  Y >. ( 1st  |`  ( A  X.  { Y } ) ) X ) )
75, 6mpan2 682 . . . . . 6  |-  ( X  e.  A  ->  ( X `' ( 1st  |`  ( A  X.  { Y }
) ) <. X ,  Y >. 
<-> 
<. X ,  Y >. ( 1st  |`  ( A  X.  { Y } ) ) X ) )
8 brresg 5131 . . . . . 6  |-  ( X  e.  A  ->  ( <. X ,  Y >. ( 1st  |`  ( A  X.  { Y } ) ) X  <->  ( <. X ,  Y >. 1st X  /\  <. X ,  Y >.  e.  ( A  X.  { Y } ) ) ) )
97, 8bitrd 261 . . . . 5  |-  ( X  e.  A  ->  ( X `' ( 1st  |`  ( A  X.  { Y }
) ) <. X ,  Y >. 
<->  ( <. X ,  Y >. 1st X  /\  <. X ,  Y >.  e.  ( A  X.  { Y } ) ) ) )
109adantr 471 . . . 4  |-  ( ( X  e.  A  /\  Y  e.  V )  ->  ( X `' ( 1st  |`  ( A  X.  { Y } ) ) <. X ,  Y >.  <-> 
( <. X ,  Y >. 1st X  /\  <. X ,  Y >.  e.  ( A  X.  { Y } ) ) ) )
11 opelxp 4882 . . . . . 6  |-  ( <. X ,  Y >.  e.  ( A  X.  { Y } )  <->  ( X  e.  A  /\  Y  e. 
{ Y } ) )
1211anbi2i 705 . . . . 5  |-  ( (
<. X ,  Y >. 1st X  /\  <. X ,  Y >.  e.  ( A  X.  { Y }
) )  <->  ( <. X ,  Y >. 1st X  /\  ( X  e.  A  /\  Y  e.  { Y } ) ) )
13 br1steqg 30464 . . . . . . 7  |-  ( ( X  e.  A  /\  Y  e.  V  /\  X  e.  A )  ->  ( <. X ,  Y >. 1st X  <->  X  =  X ) )
14133anidm13 1334 . . . . . 6  |-  ( ( X  e.  A  /\  Y  e.  V )  ->  ( <. X ,  Y >. 1st X  <->  X  =  X ) )
1514anbi1d 716 . . . . 5  |-  ( ( X  e.  A  /\  Y  e.  V )  ->  ( ( <. X ,  Y >. 1st X  /\  ( X  e.  A  /\  Y  e.  { Y } ) )  <->  ( X  =  X  /\  ( X  e.  A  /\  Y  e.  { Y } ) ) ) )
1612, 15syl5bb 265 . . . 4  |-  ( ( X  e.  A  /\  Y  e.  V )  ->  ( ( <. X ,  Y >. 1st X  /\  <. X ,  Y >.  e.  ( A  X.  { Y } ) )  <->  ( X  =  X  /\  ( X  e.  A  /\  Y  e.  { Y } ) ) ) )
1710, 16bitrd 261 . . 3  |-  ( ( X  e.  A  /\  Y  e.  V )  ->  ( X `' ( 1st  |`  ( A  X.  { Y } ) ) <. X ,  Y >.  <-> 
( X  =  X  /\  ( X  e.  A  /\  Y  e. 
{ Y } ) ) ) )
184, 17mpbird 240 . 2  |-  ( ( X  e.  A  /\  Y  e.  V )  ->  X `' ( 1st  |`  ( A  X.  { Y } ) ) <. X ,  Y >. )
19 1stconst 6910 . . . . 5  |-  ( Y  e.  V  ->  ( 1st  |`  ( A  X.  { Y } ) ) : ( A  X.  { Y } ) -1-1-onto-> A )
20 f1ocnv 5848 . . . . 5  |-  ( ( 1st  |`  ( A  X.  { Y } ) ) : ( A  X.  { Y }
)
-1-1-onto-> A  ->  `' ( 1st  |`  ( A  X.  { Y } ) ) : A -1-1-onto-> ( A  X.  { Y } ) )
21 f1ofn 5837 . . . . 5  |-  ( `' ( 1st  |`  ( A  X.  { Y }
) ) : A -1-1-onto-> ( A  X.  { Y }
)  ->  `' ( 1st  |`  ( A  X.  { Y } ) )  Fn  A )
2219, 20, 213syl 18 . . . 4  |-  ( Y  e.  V  ->  `' ( 1st  |`  ( A  X.  { Y } ) )  Fn  A )
2322adantl 472 . . 3  |-  ( ( X  e.  A  /\  Y  e.  V )  ->  `' ( 1st  |`  ( A  X.  { Y }
) )  Fn  A
)
24 simpl 463 . . 3  |-  ( ( X  e.  A  /\  Y  e.  V )  ->  X  e.  A )
25 fnbrfvb 5927 . . 3  |-  ( ( `' ( 1st  |`  ( A  X.  { Y }
) )  Fn  A  /\  X  e.  A
)  ->  ( ( `' ( 1st  |`  ( A  X.  { Y }
) ) `  X
)  =  <. X ,  Y >. 
<->  X `' ( 1st  |`  ( A  X.  { Y } ) ) <. X ,  Y >. ) )
2623, 24, 25syl2anc 671 . 2  |-  ( ( X  e.  A  /\  Y  e.  V )  ->  ( ( `' ( 1st  |`  ( A  X.  { Y } ) ) `  X )  =  <. X ,  Y >.  <-> 
X `' ( 1st  |`  ( A  X.  { Y } ) ) <. X ,  Y >. ) )
2718, 26mpbird 240 1  |-  ( ( X  e.  A  /\  Y  e.  V )  ->  ( `' ( 1st  |`  ( A  X.  { Y } ) ) `  X )  =  <. X ,  Y >. )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    = wceq 1454    e. wcel 1897   _Vcvv 3056   {csn 3979   <.cop 3985   class class class wbr 4415    X. cxp 4850   `'ccnv 4851    |` cres 4854    Fn wfn 5595   -1-1-onto->wf1o 5599   ` cfv 5600   1stc1st 6817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-1st 6819  df-2nd 6820
This theorem is referenced by: (None)
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