Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  br2ndeq Structured version   Unicode version

Theorem br2ndeq 25391
Description: Uniqueness condition for binary relationship over the  2nd relationship. (Contributed by Scott Fenton, 11-Apr-2014.) (Proof shortened by Mario Carneiro, 3-May-2015.)
Hypotheses
Ref Expression
br2ndeq.1  |-  A  e. 
_V
br2ndeq.2  |-  B  e. 
_V
br2ndeq.3  |-  C  e. 
_V
Assertion
Ref Expression
br2ndeq  |-  ( <. A ,  B >. 2nd C  <->  C  =  B
)

Proof of Theorem br2ndeq
StepHypRef Expression
1 br2ndeq.1 . . . 4  |-  A  e. 
_V
2 br2ndeq.2 . . . 4  |-  B  e. 
_V
31, 2op2nd 6348 . . 3  |-  ( 2nd `  <. A ,  B >. )  =  B
43eqeq1i 2442 . 2  |-  ( ( 2nd `  <. A ,  B >. )  =  C  <-> 
B  =  C )
5 fo2nd 6359 . . . 4  |-  2nd : _V -onto-> _V
6 fofn 5647 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
75, 6ax-mp 8 . . 3  |-  2nd  Fn  _V
8 opex 4419 . . 3  |-  <. A ,  B >.  e.  _V
9 fnbrfvb 5759 . . 3  |-  ( ( 2nd  Fn  _V  /\  <. A ,  B >.  e. 
_V )  ->  (
( 2nd `  <. A ,  B >. )  =  C  <->  <. A ,  B >. 2nd C ) )
107, 8, 9mp2an 654 . 2  |-  ( ( 2nd `  <. A ,  B >. )  =  C  <->  <. A ,  B >. 2nd C )
11 eqcom 2437 . 2  |-  ( B  =  C  <->  C  =  B )
124, 10, 113bitr3i 267 1  |-  ( <. A ,  B >. 2nd C  <->  C  =  B
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652    e. wcel 1725   _Vcvv 2948   <.cop 3809   class class class wbr 4204    Fn wfn 5441   -onto->wfo 5444   ` cfv 5446   2ndc2nd 6340
This theorem is referenced by:  dfrn5  25393  brtxp  25717  brpprod  25722  elfuns  25752  brimg  25774  brcup  25776  brcap  25777  brrestrict  25786
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-2nd 6342
  Copyright terms: Public domain W3C validator