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Theorem br2ndeq 29370
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 6708 . . 3  |-  ( 2nd `  <. A ,  B >. )  =  B
43eqeq1i 2389 . 2  |-  ( ( 2nd `  <. A ,  B >. )  =  C  <-> 
B  =  C )
5 fo2nd 6720 . . . 4  |-  2nd : _V -onto-> _V
6 fofn 5705 . . . 4  |-  ( 2nd
: _V -onto-> _V  ->  2nd 
Fn  _V )
75, 6ax-mp 5 . . 3  |-  2nd  Fn  _V
8 opex 4626 . . 3  |-  <. A ,  B >.  e.  _V
9 fnbrfvb 5814 . . 3  |-  ( ( 2nd  Fn  _V  /\  <. A ,  B >.  e. 
_V )  ->  (
( 2nd `  <. A ,  B >. )  =  C  <->  <. A ,  B >. 2nd C ) )
107, 8, 9mp2an 670 . 2  |-  ( ( 2nd `  <. A ,  B >. )  =  C  <->  <. A ,  B >. 2nd C )
11 eqcom 2391 . 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 1399    e. wcel 1826   _Vcvv 3034   <.cop 3950   class class class wbr 4367    Fn wfn 5491   -onto->wfo 5494   ` cfv 5496   2ndc2nd 6698
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fo 5502  df-fv 5504  df-2nd 6700
This theorem is referenced by:  dfrn5  29372  brtxp  29683  brpprod  29688  elfuns  29718  brimg  29740  brcup  29742  brcap  29743  brrestrict  29752
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