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Theorem mpt2eq123dva 5508
Description: An equality deduction for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
mpt2eq123dv.1  D
mpt2eq123dva.2  E
mpt2eq123dva.3 
C  F
Assertion
Ref Expression
mpt2eq123dva  ,  |->  C  D ,  E  |->  F
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   (,)    C(,)    D(,)    E(,)    F(,)

Proof of Theorem mpt2eq123dva
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 mpt2eq123dva.3 . . . . . 6 
C  F
21eqeq2d 2048 . . . . 5  C  F
32pm5.32da 425 . . . 4  C  F
4 mpt2eq123dva.2 . . . . . . . 8  E
54eleq2d 2104 . . . . . . 7  E
65pm5.32da 425 . . . . . 6  E
7 mpt2eq123dv.1 . . . . . . . 8  D
87eleq2d 2104 . . . . . . 7  D
98anbi1d 438 . . . . . 6  E  D  E
106, 9bitrd 177 . . . . 5  D  E
1110anbi1d 438 . . . 4  F  D  E  F
123, 11bitrd 177 . . 3  C  D  E  F
1312oprabbidv 5501 . 2  { <. <. , 
>. ,  >.  |  C }  { <. <. , 
>. ,  >.  |  D  E  F }
14 df-mpt2 5460 . 2  ,  |->  C  { <. <. ,  >. ,  >.  |  C }
15 df-mpt2 5460 . 2  D ,  E  |->  F  { <. <. ,  >. ,  >.  |  D  E  F }
1613, 14, 153eqtr4g 2094 1  ,  |->  C  D ,  E  |->  F
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242   wcel 1390   {coprab 5456    |-> cmpt2 5457
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-oprab 5459  df-mpt2 5460
This theorem is referenced by:  mpt2eq123dv  5509
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