Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpt2eq123dva Structured version   Unicode version

Theorem mpt2eq123dva 6137
 Description: An equality deduction for the maps to notation. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
mpt2eq123dv.1
mpt2eq123dva.2
mpt2eq123dva.3
Assertion
Ref Expression
mpt2eq123dva
Distinct variable groups:   ,   ,
Allowed substitution hints:   (,)   (,)   (,)   (,)   (,)   (,)

Proof of Theorem mpt2eq123dva
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 mpt2eq123dva.3 . . . . . 6
21eqeq2d 2449 . . . . 5
32pm5.32da 624 . . . 4
4 mpt2eq123dva.2 . . . . . . . 8
54eleq2d 2505 . . . . . . 7
65pm5.32da 624 . . . . . 6
7 mpt2eq123dv.1 . . . . . . . 8
87eleq2d 2505 . . . . . . 7
98anbi1d 687 . . . . . 6
106, 9bitrd 246 . . . . 5
1110anbi1d 687 . . . 4
123, 11bitrd 246 . . 3
1312oprabbidv 6130 . 2
14 df-mpt2 6088 . 2
15 df-mpt2 6088 . 2
1613, 14, 153eqtr4g 2495 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  coprab 6084   cmpt2 6085 This theorem is referenced by:  mpt2eq123dv  6138  natpropd  14175  fucpropd  14176  curfpropd  14332  hofpropd  14366 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-oprab 6087  df-mpt2 6088
 Copyright terms: Public domain W3C validator