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Theorem fcnvgreu 26135
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 4952 . . . 4  |-  ran  A  =  dom  `' A
21eleq2i 2529 . . 3  |-  ( Y  e.  ran  A  <->  Y  e.  dom  `' A )
3 fgreu 26134 . . . 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 5393 . . . . . 6  |-  `' `' A  C_  A
7 cnvssrndm 5460 . . . . . . . . . . 11  |-  `' A  C_  ( ran  A  X.  dom  A )
87sseli 3453 . . . . . . . . . 10  |-  ( q  e.  `' A  -> 
q  e.  ( ran 
A  X.  dom  A
) )
9 dfdm4 5133 . . . . . . . . . . 11  |-  dom  A  =  ran  `' A
101, 9xpeq12i 4963 . . . . . . . . . 10  |-  ( ran 
A  X.  dom  A
)  =  ( dom  `' A  X.  ran  `' A )
118, 10syl6eleq 2549 . . . . . . . . 9  |-  ( q  e.  `' A  -> 
q  e.  ( dom  `' A  X.  ran  `' A ) )
12 2nd1st 6722 . . . . . . . . 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 2459 . . . . . . 7  |-  ( q  e.  `' A  ->  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  =  U. `' { q } )
15 relcnv 5307 . . . . . . . 8  |-  Rel  `' A
16 cnvf1olem 6773 . . . . . . . . 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 3455 . . . . 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 5459 . . . . . . . . . . 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 3457 . . . . . . . . 9  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  p  e.  ( dom  A  X.  ran  A ) )
28 2nd1st 6722 . . . . . . . . 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 2459 . . . . . . 7  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  <. ( 2nd `  p ) ,  ( 1st `  p
) >.  =  U. `' { p } )
31 cnvf1olem 6773 . . . . . . . 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 1217 . . . . . 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 1217 . . . . . . . . 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 3990 . . . . . . . . . . 11  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  { p }  =  { <. ( 2nd `  q
) ,  ( 1st `  q ) >. } )
4140cnveqd 5116 . . . . . . . . . 10  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  `' { p }  =  `' { <. ( 2nd `  q
) ,  ( 1st `  q ) >. } )
4241unieqd 4202 . . . . . . . . 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 2498 . . . . . . . 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 1217 . . . . . . . . . 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 3990 . . . . . . . . . . 11  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  { q }  =  { <. ( 2nd `  p
) ,  ( 1st `  p ) >. } )
5049cnveqd 5116 . . . . . . . . . 10  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  `' { q }  =  `' { <. ( 2nd `  p
) ,  ( 1st `  p ) >. } )
5150unieqd 4202 . . . . . . . . 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 2498 . . . . . . . 8  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. )
5444, 53impbida 828 . . . . . . 7  |-  ( ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A
)  /\  q  e.  `' A )  ->  (
p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. 
<->  q  =  <. ( 2nd `  p ) ,  ( 1st `  p
) >. ) )
5554ralrimiva 2825 . . . . . 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 2466 . . . . . . . . . . 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 2826 . . . . . . . . 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 2913 . . . . . . . 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 2839 . . . . . . 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 3172 . . . . . 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 3248 . . . . 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 5802 . . . . . . 7  |-  ( 2nd `  q )  e.  _V
67 fvex 5802 . . . . . . 7  |-  ( 1st `  q )  e.  _V
6866, 67op2ndd 6691 . . . . . 6  |-  ( p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  ->  ( 2nd `  p )  =  ( 1st `  q
) )
6968eqeq2d 2465 . . . . 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 26019 . . 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 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   E!wreu 2797    C_ wss 3429   {csn 3978   <.cop 3984   U.cuni 4192    X. cxp 4939   `'ccnv 4940   dom cdm 4941   ran crn 4942   Rel wrel 4946   Fun wfun 5513   ` cfv 5519   1stc1st 6678   2ndc2nd 6679
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 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475
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 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rmo 2803  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-mpt 4453  df-id 4737  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-iota 5482  df-fun 5521  df-fn 5522  df-fv 5527  df-1st 6680  df-2nd 6681
This theorem is referenced by:  gsummpt2co  26387
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