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Theorem fcnvgreu 28265
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 4861 . . . 4  |-  ran  A  =  dom  `' A
21eleq2i 2500 . . 3  |-  ( Y  e.  ran  A  <->  Y  e.  dom  `' A )
3 fgreu 28264 . . . 4  |-  ( ( Fun  `' A  /\  Y  e.  dom  `' A
)  ->  E! q  e.  `'  A Y  =  ( 1st `  q ) )
43adantll 718 . . 3  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  Y  e.  dom  `' A )  ->  E! q  e.  `'  A Y  =  ( 1st `  q ) )
52, 4sylan2b 477 . 2  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  Y  e.  ran  A )  ->  E! q  e.  `'  A Y  =  ( 1st `  q ) )
6 cnvcnvss 5306 . . . . . 6  |-  `' `' A  C_  A
7 cnvssrndm 5373 . . . . . . . . . . 11  |-  `' A  C_  ( ran  A  X.  dom  A )
87sseli 3460 . . . . . . . . . 10  |-  ( q  e.  `' A  -> 
q  e.  ( ran 
A  X.  dom  A
) )
9 dfdm4 5043 . . . . . . . . . . 11  |-  dom  A  =  ran  `' A
101, 9xpeq12i 4872 . . . . . . . . . 10  |-  ( ran 
A  X.  dom  A
)  =  ( dom  `' A  X.  ran  `' A )
118, 10syl6eleq 2520 . . . . . . . . 9  |-  ( q  e.  `' A  -> 
q  e.  ( dom  `' A  X.  ran  `' A ) )
12 2nd1st 6849 . . . . . . . . 9  |-  ( q  e.  ( dom  `' A  X.  ran  `' A
)  ->  U. `' {
q }  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. )
1311, 12syl 17 . . . . . . . 8  |-  ( q  e.  `' A  ->  U. `' { q }  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )
1413eqcomd 2430 . . . . . . 7  |-  ( q  e.  `' A  ->  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  =  U. `' { q } )
15 relcnv 5223 . . . . . . . 8  |-  Rel  `' A
16 cnvf1olem 6902 . . . . . . . . 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 460 . . . . . . . 8  |-  ( ( Rel  `' A  /\  ( q  e.  `' A  /\  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  =  U. `' { q } ) )  ->  <. ( 2nd `  q ) ,  ( 1st `  q )
>.  e.  `' `' A
)
1815, 17mpan 674 . . . . . . 7  |-  ( ( q  e.  `' A  /\  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  =  U. `' { q } )  ->  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  e.  `' `' A )
1914, 18mpdan 672 . . . . . 6  |-  ( q  e.  `' A  ->  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  e.  `' `' A )
206, 19sseldi 3462 . . . . 5  |-  ( q  e.  `' A  ->  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  e.  A
)
2120adantl 467 . . . 4  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  q  e.  `' A )  ->  <. ( 2nd `  q ) ,  ( 1st `  q
) >.  e.  A )
22 simpll 758 . . . . . . 7  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  Rel  A )
23 simpr 462 . . . . . . 7  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  p  e.  A )
24 relssdmrn 5372 . . . . . . . . . . 11  |-  ( Rel 
A  ->  A  C_  ( dom  A  X.  ran  A
) )
2524adantr 466 . . . . . . . . . 10  |-  ( ( Rel  A  /\  Fun  `' A )  ->  A  C_  ( dom  A  X.  ran  A ) )
2625sselda 3464 . . . . . . . . 9  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  p  e.  ( dom  A  X.  ran  A ) )
27 2nd1st 6849 . . . . . . . . 9  |-  ( p  e.  ( dom  A  X.  ran  A )  ->  U. `' { p }  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )
2826, 27syl 17 . . . . . . . 8  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  U. `' { p }  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )
2928eqcomd 2430 . . . . . . 7  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  <. ( 2nd `  p ) ,  ( 1st `  p
) >.  =  U. `' { p } )
30 cnvf1olem 6902 . . . . . . . 8  |-  ( ( Rel  A  /\  (
p  e.  A  /\  <.
( 2nd `  p
) ,  ( 1st `  p ) >.  =  U. `' { p } ) )  ->  ( <. ( 2nd `  p ) ,  ( 1st `  p
) >.  e.  `' A  /\  p  =  U. `' { <. ( 2nd `  p
) ,  ( 1st `  p ) >. } ) )
3130simpld 460 . . . . . . 7  |-  ( ( Rel  A  /\  (
p  e.  A  /\  <.
( 2nd `  p
) ,  ( 1st `  p ) >.  =  U. `' { p } ) )  ->  <. ( 2nd `  p ) ,  ( 1st `  p )
>.  e.  `' A )
3222, 23, 29, 31syl12anc 1262 . . . . . 6  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  <. ( 2nd `  p ) ,  ( 1st `  p
) >.  e.  `' A
)
3315a1i 11 . . . . . . . . . 10  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  Rel  `' A )
34 simplr 760 . . . . . . . . . 10  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  q  e.  `' A
)
3514ad2antlr 731 . . . . . . . . . 10  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  -> 
<. ( 2nd `  q
) ,  ( 1st `  q ) >.  =  U. `' { q } )
3616simprd 464 . . . . . . . . . 10  |-  ( ( Rel  `' A  /\  ( q  e.  `' A  /\  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  =  U. `' { q } ) )  ->  q  =  U. `' { <. ( 2nd `  q
) ,  ( 1st `  q ) >. } )
3733, 34, 35, 36syl12anc 1262 . . . . . . . . 9  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  q  =  U. `' { <. ( 2nd `  q
) ,  ( 1st `  q ) >. } )
38 simpr 462 . . . . . . . . . . . 12  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. )
3938sneqd 4008 . . . . . . . . . . 11  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  { p }  =  { <. ( 2nd `  q
) ,  ( 1st `  q ) >. } )
4039cnveqd 5026 . . . . . . . . . 10  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  `' { p }  =  `' { <. ( 2nd `  q
) ,  ( 1st `  q ) >. } )
4140unieqd 4226 . . . . . . . . 9  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  U. `' { p }  =  U. `' { <. ( 2nd `  q
) ,  ( 1st `  q ) >. } )
4228ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  U. `' { p }  =  <. ( 2nd `  p ) ,  ( 1st `  p )
>. )
4337, 41, 423eqtr2d 2469 . . . . . . . 8  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  q  =  <. ( 2nd `  p ) ,  ( 1st `  p
) >. )
4430simprd 464 . . . . . . . . . . 11  |-  ( ( Rel  A  /\  (
p  e.  A  /\  <.
( 2nd `  p
) ,  ( 1st `  p ) >.  =  U. `' { p } ) )  ->  p  =  U. `' { <. ( 2nd `  p
) ,  ( 1st `  p ) >. } )
4522, 23, 29, 44syl12anc 1262 . . . . . . . . . 10  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  p  =  U. `' { <. ( 2nd `  p ) ,  ( 1st `  p
) >. } )
4645ad2antrr 730 . . . . . . . . 9  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  p  =  U. `' { <. ( 2nd `  p
) ,  ( 1st `  p ) >. } )
47 simpr 462 . . . . . . . . . . . 12  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  q  =  <. ( 2nd `  p ) ,  ( 1st `  p
) >. )
4847sneqd 4008 . . . . . . . . . . 11  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  { q }  =  { <. ( 2nd `  p
) ,  ( 1st `  p ) >. } )
4948cnveqd 5026 . . . . . . . . . 10  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  `' { q }  =  `' { <. ( 2nd `  p
) ,  ( 1st `  p ) >. } )
5049unieqd 4226 . . . . . . . . 9  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  U. `' { q }  =  U. `' { <. ( 2nd `  p
) ,  ( 1st `  p ) >. } )
5113ad2antlr 731 . . . . . . . . 9  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  U. `' { q }  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. )
5246, 50, 513eqtr2d 2469 . . . . . . . 8  |-  ( ( ( ( ( Rel 
A  /\  Fun  `' A
)  /\  p  e.  A )  /\  q  e.  `' A )  /\  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )  ->  p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. )
5343, 52impbida 840 . . . . . . 7  |-  ( ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A
)  /\  q  e.  `' A )  ->  (
p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. 
<->  q  =  <. ( 2nd `  p ) ,  ( 1st `  p
) >. ) )
5453ralrimiva 2839 . . . . . 6  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  A. q  e.  `'  A ( p  = 
<. ( 2nd `  q
) ,  ( 1st `  q ) >.  <->  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )
)
55 eqeq2 2437 . . . . . . . . 9  |-  ( r  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >.  ->  (
q  =  r  <->  q  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >. )
)
5655bibi2d 319 . . . . . . . 8  |-  ( r  =  <. ( 2nd `  p
) ,  ( 1st `  p ) >.  ->  (
( p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. 
<->  q  =  r )  <-> 
( p  =  <. ( 2nd `  q ) ,  ( 1st `  q
) >. 
<->  q  =  <. ( 2nd `  p ) ,  ( 1st `  p
) >. ) ) )
5756ralbidv 2864 . . . . . . 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 ) >. )
) )
5857rspcev 3182 . . . . . 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 ) )
5932, 54, 58syl2anc 665 . . . . 5  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  E. r  e.  `'  A A. q  e.  `'  A ( p  = 
<. ( 2nd `  q
) ,  ( 1st `  q ) >.  <->  q  =  r ) )
60 reu6 3260 . . . . 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 ) )
6159, 60sylibr 215 . . . 4  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  e.  A )  ->  E! q  e.  `'  A p  =  <. ( 2nd `  q ) ,  ( 1st `  q )
>. )
62 fvex 5888 . . . . . . 7  |-  ( 2nd `  q )  e.  _V
63 fvex 5888 . . . . . . 7  |-  ( 1st `  q )  e.  _V
6462, 63op2ndd 6815 . . . . . 6  |-  ( p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  ->  ( 2nd `  p )  =  ( 1st `  q
) )
6564eqeq2d 2436 . . . . 5  |-  ( p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >.  ->  ( Y  =  ( 2nd `  p )  <->  Y  =  ( 1st `  q ) ) )
6665adantl 467 . . . 4  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  p  =  <. ( 2nd `  q
) ,  ( 1st `  q ) >. )  ->  ( Y  =  ( 2nd `  p )  <-> 
Y  =  ( 1st `  q ) ) )
6721, 61, 66reuxfr4d 28112 . . 3  |-  ( ( Rel  A  /\  Fun  `' A )  ->  ( E! p  e.  A  Y  =  ( 2nd `  p )  <->  E! q  e.  `'  A Y  =  ( 1st `  q ) ) )
6867adantr 466 . 2  |-  ( ( ( Rel  A  /\  Fun  `' A )  /\  Y  e.  ran  A )  -> 
( E! p  e.  A  Y  =  ( 2nd `  p )  <-> 
E! q  e.  `'  A Y  =  ( 1st `  q ) ) )
695, 68mpbird 235 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 187    /\ wa 370    = wceq 1437    e. wcel 1868   A.wral 2775   E.wrex 2776   E!wreu 2777    C_ wss 3436   {csn 3996   <.cop 4002   U.cuni 4216    X. cxp 4848   `'ccnv 4849   dom cdm 4850   ran crn 4851   Rel wrel 4855   Fun wfun 5592   ` cfv 5598   1stc1st 6802   2ndc2nd 6803
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-iota 5562  df-fun 5600  df-fn 5601  df-fv 5606  df-1st 6804  df-2nd 6805
This theorem is referenced by:  gsummpt2co  28538
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