![]() |
Mathbox for Gino Giotto |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvmpovw2 | Structured version Visualization version GIF version |
Description: Change bound variables and domains in a maps-to function, using implicit substitution. (Contributed by GG, 14-Aug-2025.) |
Ref | Expression |
---|---|
cbvmpovw2.1 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐸 = 𝐹) |
cbvmpovw2.2 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷) |
cbvmpovw2.3 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
cbvmpovw2 | ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑤 ∈ 𝐷 ↦ 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧) | |
2 | cbvmpovw2.3 | . . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐴 = 𝐵) | |
3 | 1, 2 | eleq12d 2831 | . . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)) |
4 | simpr 484 | . . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤) | |
5 | cbvmpovw2.2 | . . . . . 6 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷) | |
6 | 4, 5 | eleq12d 2831 | . . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐶 ↔ 𝑤 ∈ 𝐷)) |
7 | 3, 6 | anbi12d 631 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷))) |
8 | cbvmpovw2.1 | . . . . 5 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → 𝐸 = 𝐹) | |
9 | 8 | eqeq2d 2744 | . . . 4 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (𝑡 = 𝐸 ↔ 𝑡 = 𝐹)) |
10 | 7, 9 | anbi12d 631 | . . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑤) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸) ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) ∧ 𝑡 = 𝐹))) |
11 | 10 | cbvoprab12v 7517 | . 2 ⊢ {〈〈𝑥, 𝑦〉, 𝑡〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)} = {〈〈𝑧, 𝑤〉, 𝑡〉 ∣ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) ∧ 𝑡 = 𝐹)} |
12 | df-mpo 7430 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = {〈〈𝑥, 𝑦〉, 𝑡〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)} | |
13 | df-mpo 7430 | . 2 ⊢ (𝑧 ∈ 𝐵, 𝑤 ∈ 𝐷 ↦ 𝐹) = {〈〈𝑧, 𝑤〉, 𝑡〉 ∣ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) ∧ 𝑡 = 𝐹)} | |
14 | 11, 12, 13 | 3eqtr4i 2771 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑤 ∈ 𝐷 ↦ 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1535 ∈ wcel 2104 {coprab 7426 ∈ cmpo 7427 |
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-oprab 7429 df-mpo 7430 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |