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Mirrors > Home > MPE Home > Th. List > dmxpOLD | Structured version Visualization version GIF version |
Description: Obsolete version of dmxp 5941 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 5694 | . . 3 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
2 | 1 | dmeqi 5917 | . 2 ⊢ dom (𝐴 × 𝐵) = dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
3 | n0 4358 | . . . . 5 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐵) | |
4 | 3 | biimpi 216 | . . . 4 ⊢ (𝐵 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐵) |
5 | 4 | ralrimivw 3147 | . . 3 ⊢ (𝐵 ≠ ∅ → ∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐵) |
6 | dmopab3 5932 | . . 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 2786 | 1 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∃wex 1775 ∈ wcel 2105 ≠ wne 2937 ∀wral 3058 ∅c0 4338 {copab 5209 × cxp 5686 dom cdm 5688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-dm 5698 |
This theorem is referenced by: (None) |
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