| Mathbox for Gino Giotto |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > cbvmpodavw2 | Structured version Visualization version GIF version | ||
| Description: Change bound variable and domains in a maps-to function. Deduction form. (Contributed by GG, 14-Aug-2025.) |
| Ref | Expression |
|---|---|
| cbvmpodavw2.1 | ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝐸 = 𝐹) |
| cbvmpodavw2.2 | ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷) |
| cbvmpodavw2.3 | ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| cbvmpodavw2 | ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑤 ∈ 𝐷 ↦ 𝐹)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | simplr 780 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧) | |
| 2 | cbvmpodavw2.3 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝐴 = 𝐵) | |
| 3 | 1, 2 | eleq12d 2856 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑥 ∈ 𝐴 ↔ 𝑧 ∈ 𝐵)) |
| 4 | simpr 489 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤) | |
| 5 | cbvmpodavw2.2 | . . . . . 6 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝐶 = 𝐷) | |
| 6 | 4, 5 | eleq12d 2856 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑦 ∈ 𝐶 ↔ 𝑤 ∈ 𝐷)) |
| 7 | 3, 6 | anbi12d 643 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ↔ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷))) |
| 8 | cbvmpodavw2.1 | . . . . 5 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝐸 = 𝐹) | |
| 9 | 8 | eqeq2d 2773 | . . . 4 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑡 = 𝐸 ↔ 𝑡 = 𝐹)) |
| 10 | 7, 9 | anbi12d 643 | . . 3 ⊢ (((𝜑 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸) ↔ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) ∧ 𝑡 = 𝐹))) |
| 11 | 10 | cbvoprab12davw 36815 | . 2 ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑡〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)} = {〈〈𝑧, 𝑤〉, 𝑡〉 ∣ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) ∧ 𝑡 = 𝐹)}) |
| 12 | df-mpo 7417 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = {〈〈𝑥, 𝑦〉, 𝑡〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐶) ∧ 𝑡 = 𝐸)} | |
| 13 | df-mpo 7417 | . 2 ⊢ (𝑧 ∈ 𝐵, 𝑤 ∈ 𝐷 ↦ 𝐹) = {〈〈𝑧, 𝑤〉, 𝑡〉 ∣ ((𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) ∧ 𝑡 = 𝐹)} | |
| 14 | 11, 12, 13 | 3eqtr4g 2822 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ 𝐸) = (𝑧 ∈ 𝐵, 𝑤 ∈ 𝐷 ↦ 𝐹)) |
| Colors of variables: wff setvar class |
| This proof depends on syntax axioms: → wi 4 ∧ wa 400 = wceq 1569 ∈ wcel 2142 {coprab 7413 ∈ cmpo 7414 |
| This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-ext 2734 |
| This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-sb 2096 df-clab 2741 df-cleq 2754 df-clel 2837 df-rab 3416 df-v 3456 df-dif 3907 df-un 3909 df-ss 3921 df-nul 4286 df-if 4487 df-sn 4589 df-pr 4591 df-op 4595 df-oprab 7416 df-mpo 7417 |
| This theorem is used by: (None) |
| Copyright terms: Public domain | W3C validator |