| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > mpteq12dfOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of mpteq12df 5228 as of 11-Nov-2024. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by Mario Carneiro, 11-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| mpteq12df.1 | ⊢ Ⅎ𝑥𝜑 |
| mpteq12df.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
| mpteq12df.3 | ⊢ (𝜑 → 𝐵 = 𝐷) |
| Ref | Expression |
|---|---|
| mpteq12dfOLD | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mpteq12df.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
| 2 | nfv 1914 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
| 3 | mpteq12df.2 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐶) | |
| 4 | 3 | eleq2d 2827 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶)) |
| 5 | mpteq12df.3 | . . . . 5 ⊢ (𝜑 → 𝐵 = 𝐷) | |
| 6 | 5 | eqeq2d 2748 | . . . 4 ⊢ (𝜑 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐷)) |
| 7 | 4, 6 | anbi12d 632 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷))) |
| 8 | 1, 2, 7 | opabbid 5208 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)}) |
| 9 | df-mpt 5226 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
| 10 | df-mpt 5226 | . 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐷) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)} | |
| 11 | 8, 9, 10 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 {copab 5205 ↦ cmpt 5225 |
| 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-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-opab 5206 df-mpt 5226 |
| This theorem is referenced by: (None) |
| Copyright terms: Public domain | W3C validator |