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Theorem br2ndeq 27720
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 6686 . . 3  |-  ( 2nd `  <. A ,  B >. )  =  B
43eqeq1i 2458 . 2  |-  ( ( 2nd `  <. A ,  B >. )  =  C  <-> 
B  =  C )
5 fo2nd 6697 . . . 4  |-  2nd : _V -onto-> _V
6 fofn 5720 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
75, 6ax-mp 5 . . 3  |-  2nd  Fn  _V
8 opex 4654 . . 3  |-  <. A ,  B >.  e.  _V
9 fnbrfvb 5831 . . 3  |-  ( ( 2nd  Fn  _V  /\  <. A ,  B >.  e. 
_V )  ->  (
( 2nd `  <. A ,  B >. )  =  C  <->  <. A ,  B >. 2nd C ) )
107, 8, 9mp2an 672 . 2  |-  ( ( 2nd `  <. A ,  B >. )  =  C  <->  <. A ,  B >. 2nd C )
11 eqcom 2460 . 2  |-  ( B  =  C  <->  C  =  B )
124, 10, 113bitr3i 275 1  |-  ( <. A ,  B >. 2nd C  <->  C  =  B
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370    e. wcel 1758   _Vcvv 3068   <.cop 3981   class class class wbr 4390    Fn wfn 5511   -onto->wfo 5514   ` cfv 5516   2ndc2nd 6676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-fo 5522  df-fv 5524  df-2nd 6678
This theorem is referenced by:  dfrn5  27722  brtxp  28045  brpprod  28050  elfuns  28080  brimg  28102  brcup  28104  brcap  28105  brrestrict  28114
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