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Mirrors > Home > MPE Home > Th. List > dmxpss | Structured version Visualization version GIF version |
Description: The domain of a Cartesian product is included in its first factor. (Contributed by NM, 19-Mar-2007.) |
Ref | Expression |
---|---|
dmxpss | ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq2 5696 | . . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅)) | |
2 | xp0 6154 | . . . . . 6 ⊢ (𝐴 × ∅) = ∅ | |
3 | 1, 2 | eqtrdi 2789 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅) |
4 | 3 | dmeqd 5903 | . . . 4 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅) |
5 | dm0 5918 | . . . 4 ⊢ dom ∅ = ∅ | |
6 | 4, 5 | eqtrdi 2789 | . . 3 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅) |
7 | 0ss 4395 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
8 | 6, 7 | eqsstrdi 4035 | . 2 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴) |
9 | dmxp 5926 | . . 3 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) | |
10 | eqimss 4039 | . . 3 ⊢ (dom (𝐴 × 𝐵) = 𝐴 → dom (𝐴 × 𝐵) ⊆ 𝐴) | |
11 | 9, 10 | syl 17 | . 2 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴) |
12 | 8, 11 | pm2.61ine 3026 | 1 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ≠ wne 2941 ⊆ wss 3947 ∅c0 4321 × cxp 5673 dom cdm 5675 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-dm 5685 |
This theorem is referenced by: rnxpss 6168 ssxpb 6170 resssxp 6266 funssxp 6743 dff3 7097 fparlem3 8095 fparlem4 8096 frxp2 8125 frxp3 8132 brdom3 10519 brdom5 10520 brdom4 10521 canthwelem 10641 pwfseqlem4 10653 uzrdgfni 13919 xptrrel 14923 rlimpm 15440 isohom 17719 ledm 18539 gsumxp 19836 dprd2d2 19906 tsmsxp 23641 dvbssntr 25399 gsumpart 32185 esum2d 33029 poimirlem3 36429 rtrclex 42301 trclexi 42304 rtrclexi 42305 cnvtrcl0 42310 dmtrcl 42311 rfovcnvf1od 42688 issmflem 45378 fvconstr 47424 fvconstrn0 47425 fvconstr2 47426 fvconst0ci 47427 fvconstdomi 47428 |
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