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Mirrors > Home > MPE Home > Th. List > dmxp | Structured version Visualization version GIF version |
Description: The domain of a Cartesian product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
dmxp | ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xp 5682 | . . 3 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
2 | 1 | dmeqi 5904 | . 2 ⊢ dom (𝐴 × 𝐵) = dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
3 | n0 4346 | . . . . 5 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐵) | |
4 | 3 | biimpi 215 | . . . 4 ⊢ (𝐵 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐵) |
5 | 4 | ralrimivw 3149 | . . 3 ⊢ (𝐵 ≠ ∅ → ∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐵) |
6 | dmopab3 5919 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐵 ↔ dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = 𝐴) | |
7 | 5, 6 | sylib 217 | . 2 ⊢ (𝐵 ≠ ∅ → dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = 𝐴) |
8 | 2, 7 | eqtrid 2783 | 1 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1780 ∈ wcel 2105 ≠ wne 2939 ∀wral 3060 ∅c0 4322 {copab 5210 × cxp 5674 dom cdm 5676 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-dm 5686 |
This theorem is referenced by: dmxpid 5929 rnxp 6169 dmxpss 6170 ssxpb 6173 relrelss 6272 unixp 6281 xpexr2 7914 xpexcnv 7915 frxp 8117 mpocurryd 8260 fodomr 9134 nqerf 10931 dmtrclfv 14972 pwsbas 17440 pwsle 17445 imasaddfnlem 17481 imasvscafn 17490 efgrcl 19631 frlmip 21643 txindislem 23457 metustexhalf 24385 rrxip 25238 dveq0 25853 dv11cn 25854 noxp1o 27509 noextendseq 27513 bdayfo 27523 noetasuplem2 27580 noetasuplem4 27582 noetainflem2 27584 noetainflem4 27586 mbfmcst 33722 eulerpartlemt 33834 0rrv 33914 curf 36930 curunc 36934 ismgmOLD 37182 diophrw 41960 onnog 42643 onnobdayg 42644 bdaybndbday 42646 |
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