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| Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvmptdavw2 | Structured version Visualization version GIF version | ||
| Description: Change bound variable and domain in a maps-to function. Deduction form. (Contributed by GG, 14-Aug-2025.) |
| Ref | Expression |
|---|---|
| cbvmptdavw2.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷) |
| cbvmptdavw2.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| cbvmptdavw2 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eleq1w 2823 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
| 2 | 1 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) |
| 3 | cbvmptdavw2.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) | |
| 4 | 3 | eleq2d 2826 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
| 5 | 2, 4 | bitrd 279 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
| 6 | cbvmptdavw2.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐶 = 𝐷) | |
| 7 | 6 | eqeq2d 2747 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 = 𝐶 ↔ 𝑡 = 𝐷)) |
| 8 | 5, 7 | anbi12d 632 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷))) |
| 9 | 8 | cbvopab1davw 36243 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶)} = {〈𝑦, 𝑡〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷)}) |
| 10 | df-mpt 5224 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶)} | |
| 11 | df-mpt 5224 | . 2 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = {〈𝑦, 𝑡〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷)} | |
| 12 | 9, 10, 11 | 3eqtr4g 2801 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐷)) |
| 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) |
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