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Theorem br2ndeq 25156
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 6296 . . 3  |-  ( 2nd `  <. A ,  B >. )  =  B
43eqeq1i 2395 . 2  |-  ( ( 2nd `  <. A ,  B >. )  =  C  <-> 
B  =  C )
5 fo2nd 6307 . . . 4  |-  2nd : _V -onto-> _V
6 fofn 5596 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
75, 6ax-mp 8 . . 3  |-  2nd  Fn  _V
8 opex 4369 . . 3  |-  <. A ,  B >.  e.  _V
9 fnbrfvb 5707 . . 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 2390 . 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 1649    e. wcel 1717   _Vcvv 2900   <.cop 3761   class class class wbr 4154    Fn wfn 5390   -onto->wfo 5393   ` cfv 5395   2ndc2nd 6288
This theorem is referenced by:  dfrn5  25158  brtxp  25445  brpprod  25450  elfuns  25479  brimg  25501  brcup  25503  brcap  25504  brrestrict  25513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-mpt 4210  df-id 4440  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-fo 5401  df-fv 5403  df-2nd 6290
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