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Theorem mpteq12f 3828
Description: An equality theorem for the maps to notation. (Contributed by Mario Carneiro, 16-Dec-2013.)
Assertion
Ref Expression
mpteq12f  C  D  |->  C  |->  D

Proof of Theorem mpteq12f
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 nfa1 1431 . . . 4  F/  C
2 nfra1 2349 . . . 4  F/  D
31, 2nfan 1454 . . 3  F/  C  D
4 nfv 1418 . . 3  F/  C  D
5 rsp 2363 . . . . . . 7  D  D
65imp 115 . . . . . 6  D  D
76eqeq2d 2048 . . . . 5  D  D
87pm5.32da 425 . . . 4  D  D
9 sp 1398 . . . . . 6  C  C
109eleq2d 2104 . . . . 5  C  C
1110anbi1d 438 . . . 4  C  D  C  D
128, 11sylan9bbr 436 . . 3  C  D  C  D
133, 4, 12opabbid 3813 . 2  C  D  { <. ,  >.  |  }  { <. ,  >.  |  C  D }
14 df-mpt 3811 . 2  |->  { <. ,  >.  |  }
15 df-mpt 3811 . 2  C  |->  D  { <. ,  >.  |  C  D }
1613, 14, 153eqtr4g 2094 1  C  D  |->  C  |->  D
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97  wal 1240   wceq 1242   wcel 1390  wral 2300   {copab 3808    |-> cmpt 3809
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-ral 2305  df-opab 3810  df-mpt 3811
This theorem is referenced by:  mpteq12dva  3829  mpteq12  3831  mpteq2ia  3834  mpteq2da  3837
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