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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 5697 | . . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅)) | |
2 | xp0 6157 | . . . . . 6 ⊢ (𝐴 × ∅) = ∅ | |
3 | 1, 2 | eqtrdi 2788 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅) |
4 | 3 | dmeqd 5905 | . . . 4 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅) |
5 | dm0 5920 | . . . 4 ⊢ dom ∅ = ∅ | |
6 | 4, 5 | eqtrdi 2788 | . . 3 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅) |
7 | 0ss 4396 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
8 | 6, 7 | eqsstrdi 4036 | . 2 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴) |
9 | dmxp 5928 | . . 3 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) | |
10 | eqimss 4040 | . . 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 3025 | 1 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ≠ wne 2940 ⊆ wss 3948 ∅c0 4322 × 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 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rab 3433 df-v 3476 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-rel 5683 df-cnv 5684 df-dm 5686 |
This theorem is referenced by: rnxpss 6171 ssxpb 6173 resssxp 6269 funssxp 6746 dff3 7101 fparlem3 8099 fparlem4 8100 frxp2 8129 frxp3 8136 brdom3 10522 brdom5 10523 brdom4 10524 canthwelem 10644 pwfseqlem4 10656 uzrdgfni 13922 xptrrel 14926 rlimpm 15443 isohom 17722 ledm 18542 gsumxp 19843 dprd2d2 19913 tsmsxp 23658 dvbssntr 25416 gsumpart 32202 esum2d 33086 poimirlem3 36486 rtrclex 42358 trclexi 42361 rtrclexi 42362 cnvtrcl0 42367 dmtrcl 42368 rfovcnvf1od 42745 issmflem 45433 fvconstr 47512 fvconstrn0 47513 fvconstr2 47514 fvconst0ci 47515 fvconstdomi 47516 |
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