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| Mirrors > Home > MPE Home > Th. List > mpteq12f | Structured version Visualization version GIF version | ||
| Description: An equality theorem for the maps-to notation. (Contributed by Mario Carneiro, 16-Dec-2013.) |
| Ref | Expression |
|---|---|
| mpteq12f | ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfa1 2151 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 𝐴 = 𝐶 | |
| 2 | nfra1 3284 | . . . 4 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 = 𝐷 | |
| 3 | 1, 2 | nfan 1899 | . . 3 ⊢ Ⅎ𝑥(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) |
| 4 | nfv 1914 | . . 3 ⊢ Ⅎ𝑦(∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) | |
| 5 | rspa 3248 | . . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 = 𝐷 ∧ 𝑥 ∈ 𝐴) → 𝐵 = 𝐷) | |
| 6 | 5 | eqeq2d 2748 | . . . . 5 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 = 𝐷 ∧ 𝑥 ∈ 𝐴) → (𝑦 = 𝐵 ↔ 𝑦 = 𝐷)) |
| 7 | 6 | pm5.32da 579 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 = 𝐷 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷))) |
| 8 | sp 2183 | . . . . . 6 ⊢ (∀𝑥 𝐴 = 𝐶 → 𝐴 = 𝐶) | |
| 9 | 8 | eleq2d 2827 | . . . . 5 ⊢ (∀𝑥 𝐴 = 𝐶 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐶)) |
| 10 | 9 | anbi1d 631 | . . . 4 ⊢ (∀𝑥 𝐴 = 𝐶 → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐷) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷))) |
| 11 | 7, 10 | sylan9bbr 510 | . . 3 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → ((𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵) ↔ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷))) |
| 12 | 3, 4, 11 | opabbid 5208 | . 2 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)}) |
| 13 | df-mpt 5226 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
| 14 | df-mpt 5226 | . 2 ⊢ (𝑥 ∈ 𝐶 ↦ 𝐷) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐶 ∧ 𝑦 = 𝐷)} | |
| 15 | 12, 13, 14 | 3eqtr4g 2802 | 1 ⊢ ((∀𝑥 𝐴 = 𝐶 ∧ ∀𝑥 ∈ 𝐴 𝐵 = 𝐷) → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐶 ↦ 𝐷)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 = wceq 1540 ∈ wcel 2108 ∀wral 3061 {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-10 2141 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-opab 5206 df-mpt 5226 |
| This theorem is referenced by: mpteq12dvaOLD 5232 mpteq12 5234 mpteq2daOLD 5241 mpteq2iaOLD 5246 esumeq12dvaf 34032 refsum2cnlem1 45042 mpteq1dfOLD 45242 mpteq12daOLD 45249 smfinflem 46832 |
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