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Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvmptvw2 | Structured version Visualization version GIF version |
Description: Change bound variable and domain in a maps-to function, using implicit substitution. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbvmptvw2.1 | ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) |
cbvmptvw2.2 | ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
cbvmptvw2 | ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1w 2820 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)) | |
2 | cbvmptvw2.2 | . . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐴 = 𝐵) | |
3 | 2 | eleq2d 2823 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝑦 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
4 | 1, 3 | bitrd 279 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
5 | cbvmptvw2.1 | . . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷) | |
6 | 5 | eqeq2d 2744 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝑡 = 𝐶 ↔ 𝑡 = 𝐷)) |
7 | 4, 6 | anbi12d 631 | . . 3 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶) ↔ (𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷))) |
8 | 7 | cbvopab1v 5228 | . 2 ⊢ {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶)} = {〈𝑦, 𝑡〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷)} |
9 | df-mpt 5233 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑡 = 𝐶)} | |
10 | df-mpt 5233 | . 2 ⊢ (𝑦 ∈ 𝐵 ↦ 𝐷) = {〈𝑦, 𝑡〉 ∣ (𝑦 ∈ 𝐵 ∧ 𝑡 = 𝐷)} | |
11 | 8, 9, 10 | 3eqtr4i 2771 | 1 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐵 ↦ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ 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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