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Theorem br1steq 24130
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 6128 . . 3  |-  ( 1st `  <. A ,  B >. )  =  A
43eqeq1i 2290 . 2  |-  ( ( 1st `  <. A ,  B >. )  =  C  <-> 
A  =  C )
5 fo1st 6139 . . . 4  |-  1st : _V -onto-> _V
6 fofn 5453 . . . 4  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
75, 6ax-mp 8 . . 3  |-  1st  Fn  _V
8 opex 4237 . . 3  |-  <. A ,  B >.  e.  _V
9 fnbrfvb 5563 . . 3  |-  ( ( 1st  Fn  _V  /\  <. A ,  B >.  e. 
_V )  ->  (
( 1st `  <. A ,  B >. )  =  C  <->  <. A ,  B >. 1st C ) )
107, 8, 9mp2an 653 . 2  |-  ( ( 1st `  <. A ,  B >. )  =  C  <->  <. A ,  B >. 1st C )
11 eqcom 2285 . 2  |-  ( A  =  C  <->  C  =  A )
124, 10, 113bitr3i 266 1  |-  ( <. A ,  B >. 1st C  <->  C  =  A
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   _Vcvv 2788   <.cop 3643   class class class wbr 4023    Fn wfn 5250   -onto->wfo 5253   ` cfv 5255   1stc1st 6120
This theorem is referenced by:  dfdm5  24132  brtxp  24420  brpprod  24425  elfuns  24454  brimg  24476  brcup  24478  brcap  24479  brrestrict  24487
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122
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