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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 5705 | . . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅)) | |
| 2 | xp0 6177 | . . . . . 6 ⊢ (𝐴 × ∅) = ∅ | |
| 3 | 1, 2 | eqtrdi 2792 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅) | 
| 4 | 3 | dmeqd 5915 | . . . 4 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅) | 
| 5 | dm0 5930 | . . . 4 ⊢ dom ∅ = ∅ | |
| 6 | 4, 5 | eqtrdi 2792 | . . 3 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅) | 
| 7 | 0ss 4399 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
| 8 | 6, 7 | eqsstrdi 4027 | . 2 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴) | 
| 9 | dmxp 5938 | . . 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 3024 | 1 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ≠ wne 2939 ⊆ wss 3950 ∅c0 4332 × cxp 5682 dom cdm 5684 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-dm 5694 | 
| This theorem is referenced by: rnxpss 6191 ssxpb 6193 resssxp 6289 funssxp 6763 dff3 7119 fparlem3 8140 fparlem4 8141 frxp2 8170 frxp3 8177 brdom3 10569 brdom5 10570 brdom4 10571 canthwelem 10691 pwfseqlem4 10703 uzrdgfni 14000 xptrrel 15020 rlimpm 15537 isohom 17821 ledm 18636 gsumxp 19995 dprd2d2 20065 tsmsxp 24164 dvbssntr 25936 noseqrdgfn 28313 gsumpart 33061 esum2d 34095 poimirlem3 37631 rtrclex 43635 trclexi 43638 rtrclexi 43639 cnvtrcl0 43644 dmtrcl 43645 rfovcnvf1od 44022 issmflem 46747 fvconstr 48770 fvconstrn0 48771 fvconstr2 48772 fvconst0ci 48796 fvconstdomi 48797 | 
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