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Theorem mapco2g 26723
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 7030 . . . 4  |-  ( A  e.  ( B  ^m  C )  ->  A : C --> B )
2 fco 5592 . . . 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 3625 . . . . 5  |-  ( A  e.  ( B  ^m  C )  ->  -.  ( B  ^m  C )  =  (/) )
6 reldmmap 7019 . . . . . 6  |-  Rel  dom  ^m
76ovprc1 6101 . . . . 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 7023 . . 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 1652    e. wcel 1725   _Vcvv 2948   (/)c0 3620    o. ccom 4874   -->wf 5442  (class class class)co 6073    ^m cmap 7010
This theorem is referenced by:  mapco2  26724  eldioph2  26774
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-map 7012
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