Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fgreu Structured version   Visualization version   Unicode version
Theorem fgreu 29471
Description: Exactly one point of a function's graph has a given first element. (Contributed by Thierry Arnoux, 1-Apr-2018.)
Assertion
Ref Expression
fgreu |- ( ( Fun F /\ X e. dom F ) -> E! p e. F X = ( 1st ` p ) )
Distinct variable groups:   F, p   X, p
Proof of Theorem fgreu
Dummy variable q is distinct from all other variables.
StepHypRef Expression
1 funfvop 6329 . . 3 |- ( ( Fun F /\ X e. dom F ) -> <. X , ( F ` X ) >. e. F )
2 simplll 798 . . . . . . . 8 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ X = ( 1st ` p ) ) -> Fun F )
3 funrel 5905 . . . . . . . 8 |- ( Fun F -> Rel F )
42, 3syl 17 . . . . . . 7 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ X = ( 1st ` p ) ) -> Rel F )
5 simplr 792 . . . . . . 7 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ X = ( 1st ` p ) ) -> p e. F )
6 1st2nd 7214 . . . . . . 7 |- ( ( Rel F /\ p e. F ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
74, 5, 6syl2anc 693 . . . . . 6 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ X = ( 1st ` p ) ) -> p = <. ( 1st ` p ) , ( 2nd ` p ) >. )
8 simpr 477 . . . . . . 7 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ X = ( 1st ` p ) ) -> X = ( 1st ` p ) )
9 simpllr 799 . . . . . . . 8 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ X = ( 1st ` p ) ) -> X e. dom F )
108opeq1d 4408 . . . . . . . . . 10 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ X = ( 1st ` p ) ) -> <. X , ( 2nd ` p ) >. = <. ( 1st ` p ) , ( 2nd ` p ) >. )
117, 10eqtr4d 2659 . . . . . . . . 9 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ X = ( 1st ` p ) ) -> p = <. X , ( 2nd ` p ) >. )
1211, 5eqeltrrd 2702 . . . . . . . 8 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ X = ( 1st ` p ) ) -> <. X , ( 2nd ` p ) >. e. F )
13 funopfvb 6239 . . . . . . . . 9 |- ( ( Fun F /\ X e. dom F ) -> ( ( F ` X ) = ( 2nd ` p ) <-> <. X , ( 2nd ` p ) >. e. F ) )
1413biimpar 502 . . . . . . . 8 |- ( ( ( Fun F /\ X e. dom F ) /\ <. X , ( 2nd ` p ) >. e. F ) -> ( F ` X ) = ( 2nd ` p ) )
152, 9, 12, 14syl21anc 1325 . . . . . . 7 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ X = ( 1st ` p ) ) -> ( F ` X ) = ( 2nd ` p ) )
168, 15opeq12d 4410 . . . . . 6 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ X = ( 1st ` p ) ) -> <. X , ( F ` X ) >. = <. ( 1st ` p ) , ( 2nd ` p ) >. )
177, 16eqtr4d 2659 . . . . 5 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ X = ( 1st ` p ) ) -> p = <. X , ( F ` X ) >. )
18 simpr 477 . . . . . . 7 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ p = <. X , ( F ` X ) >. ) -> p = <. X , ( F ` X ) >. )
1918fveq2d 6195 . . . . . 6 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ p = <. X , ( F ` X ) >. ) -> ( 1st ` p ) = ( 1st ` <. X , ( F ` X ) >. ) )
20 fvex 6201 . . . . . . . 8 |- ( F ` X ) e. _V
21 op1stg 7180 . . . . . . . 8 |- ( ( X e. dom F /\ ( F ` X ) e. _V ) -> ( 1st ` <. X , ( F ` X ) >. ) = X )
2220, 21mpan2 707 . . . . . . 7 |- ( X e. dom F -> ( 1st ` <. X , ( F ` X ) >. ) = X )
2322ad3antlr 767 . . . . . 6 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ p = <. X , ( F ` X ) >. ) -> ( 1st ` <. X , ( F ` X ) >. ) = X )
2419, 23eqtr2d 2657 . . . . 5 |- ( ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) /\ p = <. X , ( F ` X ) >. ) -> X = ( 1st ` p ) )
2517, 24impbida 877 . . . 4 |- ( ( ( Fun F /\ X e. dom F ) /\ p e. F ) -> ( X = ( 1st ` p ) <-> p = <. X , ( F ` X ) >. ) )
2625ralrimiva 2966 . . 3 |- ( ( Fun F /\ X e. dom F ) -> A. p e. F ( X = ( 1st ` p ) <-> p = <. X , ( F ` X ) >. ) )
27 eqeq2 2633 . . . . . 6 |- ( q = <. X , ( F ` X ) >. -> ( p = q <-> p = <. X , ( F ` X ) >. ) )
2827bibi2d 332 . . . . 5 |- ( q = <. X , ( F ` X ) >. -> ( ( X = ( 1st ` p ) <-> p = q ) <-> ( X = ( 1st ` p ) <-> p = <. X , ( F ` X ) >. ) ) )
2928ralbidv 2986 . . . 4 |- ( q = <. X , ( F ` X ) >. -> ( A. p e. F ( X = ( 1st ` p ) <-> p = q ) <-> A. p e. F ( X = ( 1st ` p ) <-> p = <. X , ( F ` X ) >. ) ) )
3029rspcev 3309 . . 3 |- ( ( <. X , ( F ` X ) >. e. F /\ A. p e. F ( X = ( 1st ` p ) <-> p = <. X , ( F ` X ) >. ) ) -> E. q e. F A. p e. F ( X = ( 1st ` p ) <-> p = q ) )
311, 26, 30syl2anc 693 . 2 |- ( ( Fun F /\ X e. dom F ) -> E. q e. F A. p e. F ( X = ( 1st ` p ) <-> p = q ) )
32 reu6 3395 . 2 |- ( E! p e. F X = ( 1st ` p ) <-> E. q e. F A. p e. F ( X = ( 1st ` p ) <-> p = q ) )
3331, 32sylibr 224 1 |- ( ( Fun F /\ X e. dom F ) -> E! p e. F X = ( 1st ` p ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   E!wreu 2914   _Vcvv 3200   <.cop 4183   dom cdm 5114   Rel wrel 5119   Fun wfun 5882   ` cfv 5888   1stc1st 7166   2ndc2nd 7167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-1st 7168  df-2nd 7169
This theorem is referenced by:  fcnvgreu  29472
  Copyright terms: Public domain W3C validator