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Theorem br1steq 24201
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 6144 . . 3  |-  ( 1st `  <. A ,  B >. )  =  A
43eqeq1i 2303 . 2  |-  ( ( 1st `  <. A ,  B >. )  =  C  <-> 
A  =  C )
5 fo1st 6155 . . . 4  |-  1st : _V -onto-> _V
6 fofn 5469 . . . 4  |-  ( 1st
: _V -onto-> _V  ->  1st 
Fn  _V )
75, 6ax-mp 8 . . 3  |-  1st  Fn  _V
8 opex 4253 . . 3  |-  <. A ,  B >.  e.  _V
9 fnbrfvb 5579 . . 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 2298 . 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 1632    e. wcel 1696   _Vcvv 2801   <.cop 3656   class class class wbr 4039    Fn wfn 5266   -onto->wfo 5269   ` cfv 5271   1stc1st 6136
This theorem is referenced by:  dfdm5  24203  brtxp  24491  brpprod  24496  elfuns  24525  brimg  24547  brcup  24549  brcap  24550  brrestrict  24559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138
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