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| Mirrors > Home > MPE Home > Th. List > mpoeq123dva | Structured version Visualization version GIF version | ||
| Description: An equality deduction for the maps-to notation. (Contributed by Mario Carneiro, 26-Jan-2017.) |
| Ref | Expression |
|---|---|
| mpoeq123dv.1 | ⊢ (𝜑 → 𝐴 = 𝐷) |
| mpoeq123dva.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐸) |
| mpoeq123dva.3 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 = 𝐹) |
| Ref | Expression |
|---|---|
| mpoeq123dva | ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mpoeq123dva.3 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → 𝐶 = 𝐹) | |
| 2 | 1 | eqeq2d 2748 | . . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝑧 = 𝐶 ↔ 𝑧 = 𝐹)) |
| 3 | 2 | pm5.32da 579 | . . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐹))) |
| 4 | mpoeq123dva.2 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐸) | |
| 5 | 4 | eleq2d 2827 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝐵 ↔ 𝑦 ∈ 𝐸)) |
| 6 | 5 | pm5.32da 579 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐸))) |
| 7 | mpoeq123dv.1 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 = 𝐷) | |
| 8 | 7 | eleq2d 2827 | . . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐷)) |
| 9 | 8 | anbi1d 631 | . . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐸) ↔ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸))) |
| 10 | 6, 9 | bitrd 279 | . . . . 5 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸))) |
| 11 | 10 | anbi1d 631 | . . . 4 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐹) ↔ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹))) |
| 12 | 3, 11 | bitrd 279 | . . 3 ⊢ (𝜑 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹))) |
| 13 | 12 | oprabbidv 7499 | . 2 ⊢ (𝜑 → {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)}) |
| 14 | df-mpo 7436 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 = 𝐶)} | |
| 15 | df-mpo 7436 | . 2 ⊢ (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹) = {〈〈𝑥, 𝑦〉, 𝑧〉 ∣ ((𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝐸) ∧ 𝑧 = 𝐹)} | |
| 16 | 13, 14, 15 | 3eqtr4g 2802 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑥 ∈ 𝐷, 𝑦 ∈ 𝐸 ↦ 𝐹)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 {coprab 7432 ∈ cmpo 7433 |
| 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-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-oprab 7435 df-mpo 7436 |
| This theorem is referenced by: mpoeq123dv 7508 natpropd 18024 fucpropd 18025 curfpropd 18278 hofpropd 18312 rngcifuestrc 20639 funcrngcsetc 20640 funcrngcsetcALT 20641 funcringcsetc 20674 rrxdsfi 25445 eengv 28994 elntg 28999 submat1n 33804 rrxtopnfi 46302 eenglngeehlnm 48660 |
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