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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvmptdavw | Structured version Visualization version GIF version |
Description: Change bound variable in a maps-to function. Deduction form. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbvmptdavw.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶) |
Ref | Expression |
---|---|
cbvmptdavw | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2823 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
2 | 1 | adantl 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) |
3 | cbvmptdavw.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐵 = 𝐶) | |
4 | 3 | eqeq2d 2747 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 = 𝐵 ↔ 𝑡 = 𝐶)) |
5 | 2, 4 | anbi12d 632 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑡 = 𝐶))) |
6 | 5 | cbvopab1davw 36243 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐵)} = {〈𝑦, 𝑡〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑡 = 𝐶)}) |
7 | df-mpt 5224 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐵)} | |
8 | df-mpt 5224 | . 2 ⊢ (𝑦 ∈ 𝐴 ↦ 𝐶) = {〈𝑦, 𝑡〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑡 = 𝐶)} | |
9 | 6, 7, 8 | 3eqtr4g 2801 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ 𝐶)) |
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: cbvproddavw 36259 cbvitgdavw 36260 cbvproddavw2 36275 cbvitgdavw2 36276 |
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