Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mpteq12df | Structured version Visualization version GIF version |
Description: An equality theorem for the maps to notation. (Contributed by Thierry Arnoux, 30-May-2020.) |
Ref | Expression |
---|---|
mpteq12df.0 | ⊢ Ⅎ𝑥𝜑 |
mpteq12df.1 | ⊢ Ⅎ𝑥𝐴 |
mpteq12df.2 | ⊢ Ⅎ𝑥𝐶 |
mpteq12df.3 | ⊢ (𝜑 → 𝐴 = 𝐶) |
mpteq12df.4 | ⊢ (𝜑 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
mpteq12df | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpteq12df.0 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | nfv 1843 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | mpteq12df.3 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐶) | |
4 | 3 | eleq2d 2687 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶)) |
5 | mpteq12df.4 | . . . . 5 ⊢ (𝜑 → 𝐵 = 𝐷) | |
6 | 5 | eqeq2d 2632 | . . . 4 ⊢ (𝜑 → (𝑦 = 𝐵 ↔ 𝑦 = 𝐷)) |
7 | 4, 6 | anbi12d 747 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷))) |
8 | 1, 2, 7 | opabbid 4715 | . 2 ⊢ (𝜑 → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)}) |
9 | df-mpt 4730 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
10 | df-mpt 4730 | . 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐷) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)} | |
11 | 8, 9, 10 | 3eqtr4g 2681 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 Ⅎwnfc 2751 {copab 4712 ↦ cmpt 4729 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-opab 4713 df-mpt 4730 |
This theorem is referenced by: esumrnmpt2 30130 smflimsuplem3 41028 |
Copyright terms: Public domain | W3C validator |