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Theorem prjpacp1 25230
Description: Projection of a part of a cross product. (Contributed by FL, 15-Oct-2012.)
Assertion
Ref Expression
prjpacp1  |-  ( ( B  =/=  (/)  /\  C  C_  ( A  X.  B
) )  ->  ( 1st " C )  C_  A )

Proof of Theorem prjpacp1
StepHypRef Expression
1 imass2 5065 . . 3  |-  ( C 
C_  ( A  X.  B )  ->  ( 1st " C )  C_  ( 1st " ( A  X.  B ) ) )
2 prjcp1 25187 . . . 4  |-  ( B  =/=  (/)  ->  ( 1st " ( A  X.  B
) )  =  A )
3 sseq2 3213 . . . . 5  |-  ( ( 1st " ( A  X.  B ) )  =  A  ->  (
( 1st " C
)  C_  ( 1st " ( A  X.  B
) )  <->  ( 1st " C )  C_  A
) )
43biimpd 198 . . . 4  |-  ( ( 1st " ( A  X.  B ) )  =  A  ->  (
( 1st " C
)  C_  ( 1st " ( A  X.  B
) )  ->  ( 1st " C )  C_  A ) )
52, 4syl 15 . . 3  |-  ( B  =/=  (/)  ->  ( ( 1st " C )  C_  ( 1st " ( A  X.  B ) )  ->  ( 1st " C
)  C_  A )
)
61, 5syl5com 26 . 2  |-  ( C 
C_  ( A  X.  B )  ->  ( B  =/=  (/)  ->  ( 1st " C )  C_  A
) )
76impcom 419 1  |-  ( ( B  =/=  (/)  /\  C  C_  ( A  X.  B
) )  ->  ( 1st " C )  C_  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    =/= wne 2459    C_ wss 3165   (/)c0 3468    X. cxp 4703   "cima 4708   1stc1st 6136
This theorem is referenced by:  limptlimpr2lem1  25677
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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