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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 2820 | . . . . . 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 2823 | . . . . 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 2744 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑡 = 𝐶 ↔ 𝑡 = 𝐷)) |
8 | 5, 7 | anbi12d 631 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷))) |
9 | 8 | cbvopab1davw 36207 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶)} = {〈𝑦, 𝑡〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷)}) |
10 | df-mpt 5233 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶)} | |
11 | df-mpt 5233 | . 2 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = {〈𝑦, 𝑡〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷)} | |
12 | 9, 10, 11 | 3eqtr4g 2798 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1535 ∈ wcel 2104 {copab 5211 ↦ cmpt 5232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-opab 5212 df-mpt 5233 |
This theorem is referenced by: (None) |
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