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Theorem mapco2g 29199
Description: Renaming indexes in a tuple, with sethood as antecedents. (Contributed by Stefan O'Rear, 9-Oct-2014.) (Revised by Mario Carneiro, 5-May-2015.)
Assertion
Ref Expression
mapco2g  |-  ( ( E  e.  _V  /\  A  e.  ( B  ^m  C )  /\  D : E --> C )  -> 
( A  o.  D
)  e.  ( B  ^m  E ) )

Proof of Theorem mapco2g
StepHypRef Expression
1 elmapi 7345 . . . 4  |-  ( A  e.  ( B  ^m  C )  ->  A : C --> B )
2 fco 5677 . . . 4  |-  ( ( A : C --> B  /\  D : E --> C )  ->  ( A  o.  D ) : E --> B )
31, 2sylan 471 . . 3  |-  ( ( A  e.  ( B  ^m  C )  /\  D : E --> C )  ->  ( A  o.  D ) : E --> B )
433adant1 1006 . 2  |-  ( ( E  e.  _V  /\  A  e.  ( B  ^m  C )  /\  D : E --> C )  -> 
( A  o.  D
) : E --> B )
5 n0i 3751 . . . . 5  |-  ( A  e.  ( B  ^m  C )  ->  -.  ( B  ^m  C )  =  (/) )
6 reldmmap 7334 . . . . . 6  |-  Rel  dom  ^m
76ovprc1 6229 . . . . 5  |-  ( -.  B  e.  _V  ->  ( B  ^m  C )  =  (/) )
85, 7nsyl2 127 . . . 4  |-  ( A  e.  ( B  ^m  C )  ->  B  e.  _V )
983ad2ant2 1010 . . 3  |-  ( ( E  e.  _V  /\  A  e.  ( B  ^m  C )  /\  D : E --> C )  ->  B  e.  _V )
10 simp1 988 . . 3  |-  ( ( E  e.  _V  /\  A  e.  ( B  ^m  C )  /\  D : E --> C )  ->  E  e.  _V )
11 elmapg 7338 . . 3  |-  ( ( B  e.  _V  /\  E  e.  _V )  ->  ( ( A  o.  D )  e.  ( B  ^m  E )  <-> 
( A  o.  D
) : E --> B ) )
129, 10, 11syl2anc 661 . 2  |-  ( ( E  e.  _V  /\  A  e.  ( B  ^m  C )  /\  D : E --> C )  -> 
( ( A  o.  D )  e.  ( B  ^m  E )  <-> 
( A  o.  D
) : E --> B ) )
134, 12mpbird 232 1  |-  ( ( E  e.  _V  /\  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    <-> wb 184    /\ w3a 965    = wceq 1370    e. wcel 1758   _Vcvv 3078   (/)c0 3746    o. ccom 4953   -->wf 5523  (class class class)co 6201    ^m cmap 7325
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 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
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 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-map 7327
This theorem is referenced by:  mapco2  29200  eldioph2  29249
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