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Theorem mapco2 29175
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 29174 . 2  |-  ( ( E  e.  _V  /\  A  e.  ( B  ^m  C )  /\  D : E --> C )  -> 
( A  o.  D
)  e.  ( B  ^m  E ) )
31, 2mp3an1 1302 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 1757   _Vcvv 3054    o. ccom 4928   -->wf 5498  (class class class)co 6176    ^m cmap 7300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-ral 2797  df-rex 2798  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-id 4720  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-fv 5510  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-1st 6663  df-2nd 6664  df-map 7302
This theorem is referenced by:  diophren  29276  rabrenfdioph  29277
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