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Theorem 1st2ndbr 6169
Description: Express an element of a relation as a relationship between first and second components. (Contributed by Mario Carneiro, 22-Jun-2016.)
Assertion
Ref Expression
1st2ndbr  |-  ( ( Rel  B  /\  A  e.  B )  ->  ( 1st `  A ) B ( 2nd `  A
) )

Proof of Theorem 1st2ndbr
StepHypRef Expression
1 1st2nd 6166 . . 3  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
2 simpr 447 . . 3  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  e.  B )
31, 2eqeltrrd 2358 . 2  |-  ( ( Rel  B  /\  A  e.  B )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  e.  B )
4 df-br 4024 . 2  |-  ( ( 1st `  A ) B ( 2nd `  A
)  <->  <. ( 1st `  A
) ,  ( 2nd `  A ) >.  e.  B
)
53, 4sylibr 203 1  |-  ( ( Rel  B  /\  A  e.  B )  ->  ( 1st `  A ) B ( 2nd `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   <.cop 3643   class class class wbr 4023   Rel wrel 4694   ` cfv 5255   1stc1st 6120   2ndc2nd 6121
This theorem is referenced by:  cofuval  13756  cofu1  13758  cofu2  13760  cofucl  13762  cofuass  13763  cofulid  13764  cofurid  13765  funcres  13770  cofull  13808  cofth  13809  isnat2  13822  fuccocl  13838  fucidcl  13839  fuclid  13840  fucrid  13841  fucass  13842  fucsect  13846  fucinv  13847  invfuc  13848  fuciso  13849  natpropd  13850  fucpropd  13851  homahom  13871  homadm  13872  homacd  13873  homadmcd  13874  catciso  13939  prfval  13973  prfcl  13977  prf1st  13978  prf2nd  13979  1st2ndprf  13980  evlfcllem  13995  evlfcl  13996  curf1cl  14002  curf2cl  14005  curfcl  14006  uncf1  14010  uncf2  14011  curfuncf  14012  uncfcurf  14013  diag1cl  14016  diag2cl  14020  curf2ndf  14021  yon1cl  14037  oyon1cl  14045  yonedalem1  14046  yonedalem21  14047  yonedalem3a  14048  yonedalem4c  14051  yonedalem22  14052  yonedalem3b  14053  yonedalem3  14054  yonedainv  14055  yonffthlem  14056  yoniso  14059
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-fv 5263  df-1st 6122  df-2nd 6123
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