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Theorem mapco2g 26462
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 6976 . . . 4  |-  ( A  e.  ( B  ^m  C )  ->  A : C --> B )
2 fco 5542 . . . 4  |-  ( ( A : C --> B  /\  D : E --> C )  ->  ( A  o.  D ) : E --> B )
31, 2sylan 458 . . 3  |-  ( ( A  e.  ( B  ^m  C )  /\  D : E --> C )  ->  ( A  o.  D ) : E --> B )
433adant1 975 . 2  |-  ( ( E  e.  _V  /\  A  e.  ( B  ^m  C )  /\  D : E --> C )  -> 
( A  o.  D
) : E --> B )
5 n0i 3578 . . . . 5  |-  ( A  e.  ( B  ^m  C )  ->  -.  ( B  ^m  C )  =  (/) )
6 reldmmap 6965 . . . . . 6  |-  Rel  dom  ^m
76ovprc1 6050 . . . . 5  |-  ( -.  B  e.  _V  ->  ( B  ^m  C )  =  (/) )
85, 7nsyl2 121 . . . 4  |-  ( A  e.  ( B  ^m  C )  ->  B  e.  _V )
983ad2ant2 979 . . 3  |-  ( ( E  e.  _V  /\  A  e.  ( B  ^m  C )  /\  D : E --> C )  ->  B  e.  _V )
10 simp1 957 . . 3  |-  ( ( E  e.  _V  /\  A  e.  ( B  ^m  C )  /\  D : E --> C )  ->  E  e.  _V )
11 elmapg 6969 . . 3  |-  ( ( B  e.  _V  /\  E  e.  _V )  ->  ( ( A  o.  D )  e.  ( B  ^m  E )  <-> 
( A  o.  D
) : E --> B ) )
129, 10, 11syl2anc 643 . 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 224 1  |-  ( ( E  e.  _V  /\  A  e.  ( B  ^m  C )  /\  D : E --> C )  -> 
( A  o.  D
)  e.  ( B  ^m  E ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    = wceq 1649    e. wcel 1717   _Vcvv 2901   (/)c0 3573    o. ccom 4824   -->wf 5392  (class class class)co 6022    ^m cmap 6956
This theorem is referenced by:  mapco2  26463  eldioph2  26513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-ral 2656  df-rex 2657  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-1st 6290  df-2nd 6291  df-map 6958
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