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Theorem mpt2eq123i 5510
Description: An equality inference for the maps to notation. (Contributed by NM, 15-Jul-2013.)
Hypotheses
Ref Expression
mpt2eq123i.1 A = 𝐷
mpt2eq123i.2 B = 𝐸
mpt2eq123i.3 𝐶 = 𝐹
Assertion
Ref Expression
mpt2eq123i (x A, y B𝐶) = (x 𝐷, y 𝐸𝐹)

Proof of Theorem mpt2eq123i
StepHypRef Expression
1 mpt2eq123i.1 . . . 4 A = 𝐷
21a1i 9 . . 3 ( ⊤ → A = 𝐷)
3 mpt2eq123i.2 . . . 4 B = 𝐸
43a1i 9 . . 3 ( ⊤ → B = 𝐸)
5 mpt2eq123i.3 . . . 4 𝐶 = 𝐹
65a1i 9 . . 3 ( ⊤ → 𝐶 = 𝐹)
72, 4, 6mpt2eq123dv 5509 . 2 ( ⊤ → (x A, y B𝐶) = (x 𝐷, y 𝐸𝐹))
87trud 1251 1 (x A, y B𝐶) = (x 𝐷, y 𝐸𝐹)
Colors of variables: wff set class
Syntax hints:   = wceq 1242  wtru 1243  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:  ofmres  5705
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