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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 5698 | . . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅)) | |
2 | xp0 6158 | . . . . . 6 ⊢ (𝐴 × ∅) = ∅ | |
3 | 1, 2 | eqtrdi 2789 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅) |
4 | 3 | dmeqd 5906 | . . . 4 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅) |
5 | dm0 5921 | . . . 4 ⊢ dom ∅ = ∅ | |
6 | 4, 5 | eqtrdi 2789 | . . 3 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅) |
7 | 0ss 4397 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
8 | 6, 7 | eqsstrdi 4037 | . 2 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴) |
9 | dmxp 5929 | . . 3 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) | |
10 | eqimss 4041 | . . 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 3949 ∅c0 4323 × cxp 5675 dom cdm 5677 |
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 5300 ax-nul 5307 ax-pr 5428 |
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 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-dm 5687 |
This theorem is referenced by: rnxpss 6172 ssxpb 6174 resssxp 6270 funssxp 6747 dff3 7102 fparlem3 8100 fparlem4 8101 frxp2 8130 frxp3 8137 brdom3 10523 brdom5 10524 brdom4 10525 canthwelem 10645 pwfseqlem4 10657 uzrdgfni 13923 xptrrel 14927 rlimpm 15444 isohom 17723 ledm 18543 gsumxp 19844 dprd2d2 19914 tsmsxp 23659 dvbssntr 25417 gsumpart 32207 esum2d 33091 poimirlem3 36491 rtrclex 42368 trclexi 42371 rtrclexi 42372 cnvtrcl0 42377 dmtrcl 42378 rfovcnvf1od 42755 issmflem 45443 fvconstr 47522 fvconstrn0 47523 fvconstr2 47524 fvconst0ci 47525 fvconstdomi 47526 |
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