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Theorem 11st22nd 25148
Description: A theorem of the 1st2nd 6182 family. (Contributed by FL, 26-Oct-2007.)
Assertion
Ref Expression
11st22nd  |-  ( ( ( Rel  B  /\  Rel  dom  B  /\  Rel  ran 
B )  /\  A  e.  B )  ->  A  =  <. <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ,  <. ( 1st `  ( 2nd `  A
) ) ,  ( 2nd `  ( 2nd `  A ) ) >. >. )

Proof of Theorem 11st22nd
StepHypRef Expression
1 1st2nd 6182 . . 3  |-  ( ( Rel  B  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
213ad2antl1 1117 . 2  |-  ( ( ( Rel  B  /\  Rel  dom  B  /\  Rel  ran 
B )  /\  A  e.  B )  ->  A  =  <. ( 1st `  A
) ,  ( 2nd `  A ) >. )
3 1stdm 6183 . . . . . . . 8  |-  ( ( Rel  B  /\  A  e.  B )  ->  ( 1st `  A )  e. 
dom  B )
4 1st2nd 6182 . . . . . . . 8  |-  ( ( Rel  dom  B  /\  ( 1st `  A )  e.  dom  B )  ->  ( 1st `  A
)  =  <. ( 1st `  ( 1st `  A
) ) ,  ( 2nd `  ( 1st `  A ) ) >.
)
53, 4sylan2 460 . . . . . . 7  |-  ( ( Rel  dom  B  /\  ( Rel  B  /\  A  e.  B ) )  -> 
( 1st `  A
)  =  <. ( 1st `  ( 1st `  A
) ) ,  ( 2nd `  ( 1st `  A ) ) >.
)
65exp32 588 . . . . . 6  |-  ( Rel 
dom  B  ->  ( Rel 
B  ->  ( A  e.  B  ->  ( 1st `  A )  =  <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ) ) )
76a1i 10 . . . . 5  |-  ( Rel 
ran  B  ->  ( Rel 
dom  B  ->  ( Rel 
B  ->  ( A  e.  B  ->  ( 1st `  A )  =  <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ) ) ) )
87com13 74 . . . 4  |-  ( Rel 
B  ->  ( Rel  dom 
B  ->  ( Rel  ran 
B  ->  ( A  e.  B  ->  ( 1st `  A )  =  <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ) ) ) )
983imp1 1164 . . 3  |-  ( ( ( Rel  B  /\  Rel  dom  B  /\  Rel  ran 
B )  /\  A  e.  B )  ->  ( 1st `  A )  = 
<. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. )
10 2ndrn 6184 . . . . . . . 8  |-  ( ( Rel  B  /\  A  e.  B )  ->  ( 2nd `  A )  e. 
ran  B )
11 1st2nd 6182 . . . . . . . 8  |-  ( ( Rel  ran  B  /\  ( 2nd `  A )  e.  ran  B )  ->  ( 2nd `  A
)  =  <. ( 1st `  ( 2nd `  A
) ) ,  ( 2nd `  ( 2nd `  A ) ) >.
)
1210, 11sylan2 460 . . . . . . 7  |-  ( ( Rel  ran  B  /\  ( Rel  B  /\  A  e.  B ) )  -> 
( 2nd `  A
)  =  <. ( 1st `  ( 2nd `  A
) ) ,  ( 2nd `  ( 2nd `  A ) ) >.
)
1312exp32 588 . . . . . 6  |-  ( Rel 
ran  B  ->  ( Rel 
B  ->  ( A  e.  B  ->  ( 2nd `  A )  =  <. ( 1st `  ( 2nd `  A ) ) ,  ( 2nd `  ( 2nd `  A ) )
>. ) ) )
1413a1i 10 . . . . 5  |-  ( Rel 
dom  B  ->  ( Rel 
ran  B  ->  ( Rel 
B  ->  ( A  e.  B  ->  ( 2nd `  A )  =  <. ( 1st `  ( 2nd `  A ) ) ,  ( 2nd `  ( 2nd `  A ) )
>. ) ) ) )
1514com3r 73 . . . 4  |-  ( Rel 
B  ->  ( Rel  dom 
B  ->  ( Rel  ran 
B  ->  ( A  e.  B  ->  ( 2nd `  A )  =  <. ( 1st `  ( 2nd `  A ) ) ,  ( 2nd `  ( 2nd `  A ) )
>. ) ) ) )
16153imp1 1164 . . 3  |-  ( ( ( Rel  B  /\  Rel  dom  B  /\  Rel  ran 
B )  /\  A  e.  B )  ->  ( 2nd `  A )  = 
<. ( 1st `  ( 2nd `  A ) ) ,  ( 2nd `  ( 2nd `  A ) )
>. )
179, 16opeq12d 3820 . 2  |-  ( ( ( Rel  B  /\  Rel  dom  B  /\  Rel  ran 
B )  /\  A  e.  B )  ->  <. ( 1st `  A ) ,  ( 2nd `  A
) >.  =  <. <. ( 1st `  ( 1st `  A
) ) ,  ( 2nd `  ( 1st `  A ) ) >. ,  <. ( 1st `  ( 2nd `  A ) ) ,  ( 2nd `  ( 2nd `  A ) )
>. >. )
182, 17eqtrd 2328 1  |-  ( ( ( Rel  B  /\  Rel  dom  B  /\  Rel  ran 
B )  /\  A  e.  B )  ->  A  =  <. <. ( 1st `  ( 1st `  A ) ) ,  ( 2nd `  ( 1st `  A ) )
>. ,  <. ( 1st `  ( 2nd `  A
) ) ,  ( 2nd `  ( 2nd `  A ) ) >. >. )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   <.cop 3656   dom cdm 4705   ran crn 4706   Rel wrel 4710   ` cfv 5271   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  dedalg  25846  catded  25867
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-int 3879  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-fv 5279  df-1st 6138  df-2nd 6139
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