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Theorem br1steq 27730
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 6696 . . 3  |-  ( 1st `  <. A ,  B >. )  =  A
43eqeq1i 2461 . 2  |-  ( ( 1st `  <. A ,  B >. )  =  C  <-> 
A  =  C )
5 fo1st 6707 . . . 4  |-  1st : _V -onto-> _V
6 fofn 5731 . . . 4  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
75, 6ax-mp 5 . . 3  |-  1st  Fn  _V
8 opex 4665 . . 3  |-  <. A ,  B >.  e.  _V
9 fnbrfvb 5842 . . 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 2463 . 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 1370    e. wcel 1758   _Vcvv 3078   <.cop 3992   class class class wbr 4401    Fn wfn 5522   -onto->wfo 5525   ` cfv 5527   1stc1st 6686
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 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fo 5533  df-fv 5535  df-1st 6688
This theorem is referenced by:  dfdm5  27732  brtxp  28056  brpprod  28061  elfuns  28091  brimg  28113  brcup  28115  brcap  28116  brrestrict  28125
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