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Theorem br1steq 25156
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 6296 . . 3  |-  ( 1st `  <. A ,  B >. )  =  A
43eqeq1i 2396 . 2  |-  ( ( 1st `  <. A ,  B >. )  =  C  <-> 
A  =  C )
5 fo1st 6307 . . . 4  |-  1st : _V -onto-> _V
6 fofn 5597 . . . 4  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
75, 6ax-mp 8 . . 3  |-  1st  Fn  _V
8 opex 4370 . . 3  |-  <. A ,  B >.  e.  _V
9 fnbrfvb 5708 . . 3  |-  ( ( 1st  Fn  _V  /\  <. A ,  B >.  e. 
_V )  ->  (
( 1st `  <. A ,  B >. )  =  C  <->  <. A ,  B >. 1st C ) )
107, 8, 9mp2an 654 . 2  |-  ( ( 1st `  <. A ,  B >. )  =  C  <->  <. A ,  B >. 1st C )
11 eqcom 2391 . 2  |-  ( A  =  C  <->  C  =  A )
124, 10, 113bitr3i 267 1  |-  ( <. A ,  B >. 1st C  <->  C  =  A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649    e. wcel 1717   _Vcvv 2901   <.cop 3762   class class class wbr 4155    Fn wfn 5391   -onto->wfo 5394   ` cfv 5396   1stc1st 6288
This theorem is referenced by:  dfdm5  25158  brtxp  25446  brpprod  25451  elfuns  25480  brimg  25502  brcup  25504  brcap  25505  brrestrict  25514
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 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
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 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fo 5402  df-fv 5404  df-1st 6290
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