Users' Mathboxes Mathbox for Gino Giotto < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  cbvmptdavw2 Structured version   Visualization version   GIF version

Theorem cbvmptdavw2 36267
Description: Change bound variable and domain in a maps-to function. Deduction form. (Contributed by GG, 14-Aug-2025.)
Hypotheses
Ref Expression
cbvmptdavw2.1 ((𝜑𝑥 = 𝑦) → 𝐶 = 𝐷)
cbvmptdavw2.2 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
Assertion
Ref Expression
cbvmptdavw2 (𝜑 → (𝑥𝐴𝐶) = (𝑦𝐵𝐷))
Distinct variable groups:   𝜑,𝑥,𝑦   𝑦,𝐴   𝑥,𝐵   𝑦,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem cbvmptdavw2
Dummy variable 𝑡 is distinct from all other variables.
StepHypRef Expression
1 eleq1w 2823 . . . . . 6 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
21adantl 481 . . . . 5 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐴))
3 cbvmptdavw2.2 . . . . . 6 ((𝜑𝑥 = 𝑦) → 𝐴 = 𝐵)
43eleq2d 2826 . . . . 5 ((𝜑𝑥 = 𝑦) → (𝑦𝐴𝑦𝐵))
52, 4bitrd 279 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑥𝐴𝑦𝐵))
6 cbvmptdavw2.1 . . . . 5 ((𝜑𝑥 = 𝑦) → 𝐶 = 𝐷)
76eqeq2d 2747 . . . 4 ((𝜑𝑥 = 𝑦) → (𝑡 = 𝐶𝑡 = 𝐷))
85, 7anbi12d 632 . . 3 ((𝜑𝑥 = 𝑦) → ((𝑥𝐴𝑡 = 𝐶) ↔ (𝑦𝐵𝑡 = 𝐷)))
98cbvopab1davw 36243 . 2 (𝜑 → {⟨𝑥, 𝑡⟩ ∣ (𝑥𝐴𝑡 = 𝐶)} = {⟨𝑦, 𝑡⟩ ∣ (𝑦𝐵𝑡 = 𝐷)})
10 df-mpt 5224 . 2 (𝑥𝐴𝐶) = {⟨𝑥, 𝑡⟩ ∣ (𝑥𝐴𝑡 = 𝐶)}
11 df-mpt 5224 . 2 (𝑦𝐵𝐷) = {⟨𝑦, 𝑡⟩ ∣ (𝑦𝐵𝑡 = 𝐷)}
129, 10, 113eqtr4g 2801 1 (𝜑 → (𝑥𝐴𝐶) = (𝑦𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  {copab 5203  cmpt 5223
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-opab 5204  df-mpt 5224
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator