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Theorem relinccppr 25232
Description: A relation is included in the cross product of its projections. (Contributed by FL, 15-Oct-2012.)
Assertion
Ref Expression
relinccppr  |-  ( Rel 
A  ->  A  C_  (
( 1st " A
)  X.  ( 2nd " A ) ) )

Proof of Theorem relinccppr
StepHypRef Expression
1 prjdmn 25185 . . 3  |-  ( Rel 
A  ->  ( 1st " A )  =  dom  A )
2 prjrn 25186 . . 3  |-  ( Rel 
A  ->  ( 2nd " A )  =  ran  A )
3 xpeq12 4724 . . . 4  |-  ( ( ( 1st " A
)  =  dom  A  /\  ( 2nd " A
)  =  ran  A
)  ->  ( ( 1st " A )  X.  ( 2nd " A
) )  =  ( dom  A  X.  ran  A ) )
4 relssdmrn 5209 . . . . 5  |-  ( Rel 
A  ->  A  C_  ( dom  A  X.  ran  A
) )
5 sseq2 3213 . . . . 5  |-  ( ( ( 1st " A
)  X.  ( 2nd " A ) )  =  ( dom  A  X.  ran  A )  ->  ( A  C_  ( ( 1st " A )  X.  ( 2nd " A ) )  <-> 
A  C_  ( dom  A  X.  ran  A ) ) )
64, 5syl5ibr 212 . . . 4  |-  ( ( ( 1st " A
)  X.  ( 2nd " A ) )  =  ( dom  A  X.  ran  A )  ->  ( Rel  A  ->  A  C_  (
( 1st " A
)  X.  ( 2nd " A ) ) ) )
73, 6syl 15 . . 3  |-  ( ( ( 1st " A
)  =  dom  A  /\  ( 2nd " A
)  =  ran  A
)  ->  ( Rel  A  ->  A  C_  (
( 1st " A
)  X.  ( 2nd " A ) ) ) )
81, 2, 7syl2anc 642 . 2  |-  ( Rel 
A  ->  ( Rel  A  ->  A  C_  (
( 1st " A
)  X.  ( 2nd " A ) ) ) )
98pm2.43i 43 1  |-  ( Rel 
A  ->  A  C_  (
( 1st " A
)  X.  ( 2nd " A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    C_ wss 3165    X. cxp 4703   dom cdm 4705   ran crn 4706   "cima 4708   Rel wrel 4710   1stc1st 6136   2ndc2nd 6137
This theorem is referenced by:  limptlimpr2lem2  25678
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-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139
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