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Theorem mapco2 30249
Description: Post-composition (renaming indexes) of a mapping viewed as a point. (Contributed by Stefan O'Rear, 5-Oct-2014.) (Revised by Stefan O'Rear, 5-May-2015.)
Hypothesis
Ref Expression
mapco2.3  |-  E  e. 
_V
Assertion
Ref Expression
mapco2  |-  ( ( A  e.  ( B  ^m  C )  /\  D : E --> C )  ->  ( A  o.  D )  e.  ( B  ^m  E ) )

Proof of Theorem mapco2
StepHypRef Expression
1 mapco2.3 . 2  |-  E  e. 
_V
2 mapco2g 30248 . 2  |-  ( ( E  e.  _V  /\  A  e.  ( B  ^m  C )  /\  D : E --> C )  -> 
( A  o.  D
)  e.  ( B  ^m  E ) )
31, 2mp3an1 1311 1  |-  ( ( A  e.  ( B  ^m  C )  /\  D : E --> C )  ->  ( A  o.  D )  e.  ( B  ^m  E ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   _Vcvv 3113    o. ccom 5003   -->wf 5582  (class class class)co 6282    ^m cmap 7417
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-map 7419
This theorem is referenced by:  diophren  30349  rabrenfdioph  30350
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