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Theorem fcnvgreu 27186
Description: If the converse of a relation  A is a function, exactly one point of its graph has a given second element (that is, function value) (Contributed by Thierry Arnoux, 1-Apr-2018.)
Assertion
Ref Expression
fcnvgreu  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  Y  e.  ran  A )  ->  E! p  e.  A  Y  =  ( 2nd `  p ) )
Distinct variable groups:    A, p    Y, p

Proof of Theorem fcnvgreu
Dummy variables  q 
r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-rn 5010 . . . 4  |-  ran  A  =  dom  `' A
21eleq2i 2545 . . 3  |-  ( Y  e.  ran  A  <->  Y  e.  dom  `' A )
3 fgreu 27185 . . . 4  |-  ( ( Fun  `' A  /\  Y  e.  dom  `' A
)  ->  E! q  e.  `'  A Y  =  ( 1st `  q ) )
43adantll 713 . . 3  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  Y  e.  dom  `' A )  ->  E! q  e.  `'  A Y  =  ( 1st `  q ) )
52, 4sylan2b 475 . 2  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  Y  e.  ran  A )  ->  E! q  e.  `'  A Y  =  ( 1st `  q ) )
6 cnvcnvss 5459 . . . . . 6  |-  `' `' A  C_  A
7 cnvssrndm 5527 . . . . . . . . . . 11  |-  `' A  C_  ( ran  A  X.  dom  A )
87sseli 3500 . . . . . . . . . 10  |-  ( q  e.  `' A  -> 
q  e.  ( ran 
A  X.  dom  A
) )
9 dfdm4 5193 . . . . . . . . . . 11  |-  dom  A  =  ran  `' A
101, 9xpeq12i 5021 . . . . . . . . . 10  |-  ( ran 
A  X.  dom  A
)  =  ( dom  `' A  X.  ran  `' A )
118, 10syl6eleq 2565 . . . . . . . . 9  |-  ( q  e.  `' A  -> 
q  e.  ( dom  `' A  X.  ran  `' A ) )
12 2nd1st 6826 . . . . . . . . 9  |-  ( q  e.  ( dom  `' A  X.  ran  `' A
)  ->  U. `' {
q }  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. )
1311, 12syl 16 . . . . . . . 8  |-  ( q  e.  `' A  ->  U. `' { q }  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )
1413eqcomd 2475 . . . . . . 7  |-  ( q  e.  `' A  ->  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  =  U. `' { q } )
15 relcnv 5372 . . . . . . . 8  |-  Rel  `' A
16 cnvf1olem 6878 . . . . . . . . 9  |-  ( ( Rel  `' A  /\  ( q  e.  `' A  /\  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  =  U. `' { q } ) )  ->  ( <. ( 2nd `  q ) ,  ( 1st `  q
) >.  e.  `' `' A  /\  q  =  U. `' { <. ( 2nd `  q
) ,  ( 1st `  q ) >. } ) )
1716simpld 459 . . . . . . . 8  |-  ( ( Rel  `' A  /\  ( q  e.  `' A  /\  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  =  U. `' { q } ) )  ->  <. ( 2nd `  q ) ,  ( 1st `  q )
>.  e.  `' `' A
)
1815, 17mpan 670 . . . . . . 7  |-  ( ( q  e.  `' A  /\  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  =  U. `' { q } )  ->  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  e.  `' `' A )
1914, 18mpdan 668 . . . . . 6  |-  ( q  e.  `' A  ->  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  e.  `' `' A )
206, 19sseldi 3502 . . . . 5  |-  ( q  e.  `' A  ->  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  e.  A
)
2120adantl 466 . . . 4  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  q  e.  `' A )  ->  <. ( 2nd `  q ) ,  ( 1st `  q
) >.  e.  A )
22 simpl 457 . . . . . . . 8  |-  ( ( Rel  A  /\  Fun  `' A )  ->  Rel  A )
2322adantr 465 . . . . . . 7  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  Rel  A )
24 simpr 461 . . . . . . 7  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  p  e.  A )
25 relssdmrn 5526 . . . . . . . . . . 11  |-  ( Rel 
A  ->  A  C_  ( dom  A  X.  ran  A
) )
2622, 25syl 16 . . . . . . . . . 10  |-  ( ( Rel  A  /\  Fun  `' A )  ->  A  C_  ( dom  A  X.  ran  A ) )
2726sselda 3504 . . . . . . . . 9  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  p  e.  ( dom  A  X.  ran  A ) )
28 2nd1st 6826 . . . . . . . . 9  |-  ( p  e.  ( dom  A  X.  ran  A )  ->  U. `' { p }  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )
2927, 28syl 16 . . . . . . . 8  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  U. `' { p }  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )
3029eqcomd 2475 . . . . . . 7  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  <. ( 2nd `  p ) ,  ( 1st `  p
) >.  =  U. `' { p } )
31 cnvf1olem 6878 . . . . . . . 8  |-  ( ( Rel  A  /\  (
p  e.  A  /\  <.
( 2nd `  p
) ,  ( 1st `  p ) >.  =  U. `' { p } ) )  ->  ( <. ( 2nd `  p ) ,  ( 1st `  p
) >.  e.  `' A  /\  p  =  U. `' { <. ( 2nd `  p
) ,  ( 1st `  p ) >. } ) )
3231simpld 459 . . . . . . 7  |-  ( ( Rel  A  /\  (
p  e.  A  /\  <.
( 2nd `  p
) ,  ( 1st `  p ) >.  =  U. `' { p } ) )  ->  <. ( 2nd `  p ) ,  ( 1st `  p )
>.  e.  `' A )
3323, 24, 30, 32syl12anc 1226 . . . . . 6  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  <. ( 2nd `  p ) ,  ( 1st `  p
) >.  e.  `' A
)
3415a1i 11 . . . . . . . . . 10  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  Rel  `' A )
35 simplr 754 . . . . . . . . . 10  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  q  e.  `' A
)
3614ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  -> 
<. ( 2nd `  q
) ,  ( 1st `  q ) >.  =  U. `' { q } )
3716simprd 463 . . . . . . . . . 10  |-  ( ( Rel  `' A  /\  ( q  e.  `' A  /\  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  =  U. `' { q } ) )  ->  q  =  U. `' { <. ( 2nd `  q
) ,  ( 1st `  q ) >. } )
3834, 35, 36, 37syl12anc 1226 . . . . . . . . 9  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  q  =  U. `' { <. ( 2nd `  q
) ,  ( 1st `  q ) >. } )
39 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. )
4039sneqd 4039 . . . . . . . . . . 11  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  { p }  =  { <. ( 2nd `  q
) ,  ( 1st `  q ) >. } )
4140cnveqd 5176 . . . . . . . . . 10  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  `' { p }  =  `' { <. ( 2nd `  q
) ,  ( 1st `  q ) >. } )
4241unieqd 4255 . . . . . . . . 9  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  U. `' { p }  =  U. `' { <. ( 2nd `  q
) ,  ( 1st `  q ) >. } )
4329ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  U. `' { p }  =  <. ( 2nd `  p ) ,  ( 1st `  p )
>. )
4438, 42, 433eqtr2d 2514 . . . . . . . 8  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  q  =  <. ( 2nd `  p ) ,  ( 1st `  p
) >. )
4531simprd 463 . . . . . . . . . . 11  |-  ( ( Rel  A  /\  (
p  e.  A  /\  <.
( 2nd `  p
) ,  ( 1st `  p ) >.  =  U. `' { p } ) )  ->  p  =  U. `' { <. ( 2nd `  p
) ,  ( 1st `  p ) >. } )
4623, 24, 30, 45syl12anc 1226 . . . . . . . . . 10  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  p  =  U. `' { <. ( 2nd `  p ) ,  ( 1st `  p
) >. } )
4746ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  p  =  U. `' { <. ( 2nd `  p
) ,  ( 1st `  p ) >. } )
48 simpr 461 . . . . . . . . . . . 12  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  q  =  <. ( 2nd `  p ) ,  ( 1st `  p
) >. )
4948sneqd 4039 . . . . . . . . . . 11  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  { q }  =  { <. ( 2nd `  p
) ,  ( 1st `  p ) >. } )
5049cnveqd 5176 . . . . . . . . . 10  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  `' { q }  =  `' { <. ( 2nd `  p
) ,  ( 1st `  p ) >. } )
5150unieqd 4255 . . . . . . . . 9  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  U. `' { q }  =  U. `' { <. ( 2nd `  p
) ,  ( 1st `  p ) >. } )
5213ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  U. `' { q }  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. )
5347, 51, 523eqtr2d 2514 . . . . . . . 8  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. )
5444, 53impbida 830 . . . . . . 7  |-  ( ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A
)  /\  q  e.  `' A )  ->  (
p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. 
<->  q  =  <. ( 2nd `  p ) ,  ( 1st `  p
) >. ) )
5554ralrimiva 2878 . . . . . 6  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  A. q  e.  `'  A ( p  = 
<. ( 2nd `  q
) ,  ( 1st `  q ) >.  <->  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )
)
56 biidd 237 . . . . . . . . . . 11  |-  ( r  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >.  ->  (
p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. 
<->  p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. ) )
57 eqeq2 2482 . . . . . . . . . . 11  |-  ( r  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >.  ->  (
q  =  r  <->  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )
)
5856, 57bibi12d 321 . . . . . . . . . 10  |-  ( r  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >.  ->  (
( p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. 
<->  q  =  r )  <-> 
( p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. 
<->  q  =  <. ( 2nd `  p ) ,  ( 1st `  p
) >. ) ) )
5958ralrimivw 2879 . . . . . . . . 9  |-  ( r  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >.  ->  A. q  e.  `'  A ( ( p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  <->  q  =  r )  <->  ( p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  <->  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )
) )
6059r19.21bi 2833 . . . . . . . 8  |-  ( ( r  =  <. ( 2nd `  p ) ,  ( 1st `  p
) >.  /\  q  e.  `' A )  ->  (
( p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. 
<->  q  =  r )  <-> 
( p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. 
<->  q  =  <. ( 2nd `  p ) ,  ( 1st `  p
) >. ) ) )
6160ralbidva 2900 . . . . . . 7  |-  ( r  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >.  ->  ( A. q  e.  `'  A ( p  = 
<. ( 2nd `  q
) ,  ( 1st `  q ) >.  <->  q  =  r )  <->  A. q  e.  `'  A ( p  = 
<. ( 2nd `  q
) ,  ( 1st `  q ) >.  <->  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )
) )
6261rspcev 3214 . . . . . 6  |-  ( (
<. ( 2nd `  p
) ,  ( 1st `  p ) >.  e.  `' A  /\  A. q  e.  `'  A ( p  = 
<. ( 2nd `  q
) ,  ( 1st `  q ) >.  <->  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )
)  ->  E. r  e.  `'  A A. q  e.  `'  A ( p  = 
<. ( 2nd `  q
) ,  ( 1st `  q ) >.  <->  q  =  r ) )
6333, 55, 62syl2anc 661 . . . . 5  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  E. r  e.  `'  A A. q  e.  `'  A ( p  = 
<. ( 2nd `  q
) ,  ( 1st `  q ) >.  <->  q  =  r ) )
64 reu6 3292 . . . . 5  |-  ( E! q  e.  `'  A p  =  <. ( 2nd `  q ) ,  ( 1st `  q )
>. 
<->  E. r  e.  `'  A A. q  e.  `'  A ( p  = 
<. ( 2nd `  q
) ,  ( 1st `  q ) >.  <->  q  =  r ) )
6563, 64sylibr 212 . . . 4  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  E! q  e.  `'  A p  =  <. ( 2nd `  q ) ,  ( 1st `  q )
>. )
66 fvex 5874 . . . . . . 7  |-  ( 2nd `  q )  e.  _V
67 fvex 5874 . . . . . . 7  |-  ( 1st `  q )  e.  _V
6866, 67op2ndd 6792 . . . . . 6  |-  ( p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  ->  ( 2nd `  p )  =  ( 1st `  q
) )
6968eqeq2d 2481 . . . . 5  |-  ( p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  ->  ( Y  =  ( 2nd `  p )  <->  Y  =  ( 1st `  q ) ) )
7069adantl 466 . . . 4  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  ( Y  =  ( 2nd `  p )  <-> 
Y  =  ( 1st `  q ) ) )
7121, 65, 70reuxfr4d 27065 . . 3  |-  ( ( Rel  A  /\  Fun  `' A )  ->  ( E! p  e.  A  Y  =  ( 2nd `  p )  <->  E! q  e.  `'  A Y  =  ( 1st `  q ) ) )
7271adantr 465 . 2  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  Y  e.  ran  A )  -> 
( E! p  e.  A  Y  =  ( 2nd `  p )  <-> 
E! q  e.  `'  A Y  =  ( 1st `  q ) ) )
735, 72mpbird 232 1  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  Y  e.  ran  A )  ->  E! p  e.  A  Y  =  ( 2nd `  p ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   E!wreu 2816    C_ wss 3476   {csn 4027   <.cop 4033   U.cuni 4245    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000   Rel wrel 5004   Fun wfun 5580   ` cfv 5586   1stc1st 6779   2ndc2nd 6780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-iota 5549  df-fun 5588  df-fn 5589  df-fv 5594  df-1st 6781  df-2nd 6782
This theorem is referenced by:  gsummpt2co  27434
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