Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  br1steq Structured version   Unicode version

Theorem br1steq 29421
Description: Uniqueness condition for binary relationship over the  1st relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
Hypotheses
Ref Expression
br1steq.1  |-  A  e. 
_V
br1steq.2  |-  B  e. 
_V
br1steq.3  |-  C  e. 
_V
Assertion
Ref Expression
br1steq  |-  ( <. A ,  B >. 1st C  <->  C  =  A
)

Proof of Theorem br1steq
StepHypRef Expression
1 br1steq.1 . . . 4  |-  A  e. 
_V
2 br1steq.2 . . . 4  |-  B  e. 
_V
31, 2op1st 6807 . . 3  |-  ( 1st `  <. A ,  B >. )  =  A
43eqeq1i 2464 . 2  |-  ( ( 1st `  <. A ,  B >. )  =  C  <-> 
A  =  C )
5 fo1st 6819 . . . 4  |-  1st : _V -onto-> _V
6 fofn 5803 . . . 4  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
75, 6ax-mp 5 . . 3  |-  1st  Fn  _V
8 opex 4720 . . 3  |-  <. A ,  B >.  e.  _V
9 fnbrfvb 5913 . . 3  |-  ( ( 1st  Fn  _V  /\  <. A ,  B >.  e. 
_V )  ->  (
( 1st `  <. A ,  B >. )  =  C  <->  <. A ,  B >. 1st C ) )
107, 8, 9mp2an 672 . 2  |-  ( ( 1st `  <. A ,  B >. )  =  C  <->  <. A ,  B >. 1st C )
11 eqcom 2466 . 2  |-  ( A  =  C  <->  C  =  A )
124, 10, 113bitr3i 275 1  |-  ( <. A ,  B >. 1st C  <->  C  =  A
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1395    e. wcel 1819   _Vcvv 3109   <.cop 4038   class class class wbr 4456    Fn wfn 5589   -onto->wfo 5592   ` cfv 5594   1stc1st 6797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-fo 5600  df-fv 5602  df-1st 6799
This theorem is referenced by:  dfdm5  29423  brtxp  29735  brpprod  29740  elfuns  29770  brimg  29792  brcup  29794  brcap  29795  brrestrict  29804
  Copyright terms: Public domain W3C validator