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| Mirrors > Home > MPE Home > Th. List > dmxpOLD | Structured version Visualization version GIF version | ||
| Description: Obsolete version of dmxp 5939 as of 19-Dec-2024. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) (Proof modification is discouraged.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| dmxpOLD | ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-xp 5691 | . . 3 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
| 2 | 1 | dmeqi 5915 | . 2 ⊢ dom (𝐴 × 𝐵) = dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
| 3 | n0 4353 | . . . . 5 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐵) | |
| 4 | 3 | biimpi 216 | . . . 4 ⊢ (𝐵 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐵) |
| 5 | 4 | ralrimivw 3150 | . . 3 ⊢ (𝐵 ≠ ∅ → ∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐵) |
| 6 | dmopab3 5930 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐵 ↔ dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = 𝐴) | |
| 7 | 5, 6 | sylib 218 | . 2 ⊢ (𝐵 ≠ ∅ → dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = 𝐴) |
| 8 | 2, 7 | eqtrid 2789 | 1 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∀wral 3061 ∅c0 4333 {copab 5205 × cxp 5683 dom cdm 5685 |
| 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-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-dm 5695 |
| This theorem is referenced by: (None) |
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