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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 5710 | . . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅)) | |
2 | xp0 6180 | . . . . . 6 ⊢ (𝐴 × ∅) = ∅ | |
3 | 1, 2 | eqtrdi 2791 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅) |
4 | 3 | dmeqd 5919 | . . . 4 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅) |
5 | dm0 5934 | . . . 4 ⊢ dom ∅ = ∅ | |
6 | 4, 5 | eqtrdi 2791 | . . 3 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅) |
7 | 0ss 4406 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
8 | 6, 7 | eqsstrdi 4050 | . 2 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴) |
9 | dmxp 5942 | . . 3 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) | |
10 | eqimss 4054 | . . 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 3023 | 1 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ≠ wne 2938 ⊆ wss 3963 ∅c0 4339 × cxp 5687 dom cdm 5689 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 df-cnv 5697 df-dm 5699 |
This theorem is referenced by: rnxpss 6194 ssxpb 6196 resssxp 6292 funssxp 6765 dff3 7120 fparlem3 8138 fparlem4 8139 frxp2 8168 frxp3 8175 brdom3 10566 brdom5 10567 brdom4 10568 canthwelem 10688 pwfseqlem4 10700 uzrdgfni 13996 xptrrel 15016 rlimpm 15533 isohom 17824 ledm 18648 gsumxp 20009 dprd2d2 20079 tsmsxp 24179 dvbssntr 25950 noseqrdgfn 28327 gsumpart 33043 esum2d 34074 poimirlem3 37610 rtrclex 43607 trclexi 43610 rtrclexi 43611 cnvtrcl0 43616 dmtrcl 43617 rfovcnvf1od 43994 issmflem 46683 fvconstr 48686 fvconstrn0 48687 fvconstr2 48688 fvconst0ci 48689 fvconstdomi 48690 |
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