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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 a subclass of the first factor. (Contributed by NM, 19-Mar-2007.) |
Ref | Expression |
---|---|
dmxpss | ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | xpeq2 5053 | . . . . . 6 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅)) | |
2 | xp0 5471 | . . . . . 6 ⊢ (𝐴 × ∅) = ∅ | |
3 | 1, 2 | syl6eq 2660 | . . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅) |
4 | 3 | dmeqd 5248 | . . . 4 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = dom ∅) |
5 | dm0 5260 | . . . 4 ⊢ dom ∅ = ∅ | |
6 | 4, 5 | syl6eq 2660 | . . 3 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) = ∅) |
7 | 0ss 3924 | . . 3 ⊢ ∅ ⊆ 𝐴 | |
8 | 6, 7 | syl6eqss 3618 | . 2 ⊢ (𝐵 = ∅ → dom (𝐴 × 𝐵) ⊆ 𝐴) |
9 | dmxp 5265 | . . 3 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) | |
10 | eqimss 3620 | . . 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 2865 | 1 ⊢ dom (𝐴 × 𝐵) ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ≠ wne 2780 ⊆ wss 3540 ∅c0 3874 × cxp 5036 dom cdm 5038 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-xp 5044 df-rel 5045 df-cnv 5046 df-dm 5048 |
This theorem is referenced by: rnxpss 5485 ssxpb 5487 funssxp 5974 dff3 6280 fparlem3 7166 fparlem4 7167 brdom3 9231 brdom5 9232 brdom4 9233 canthwelem 9351 pwfseqlem4 9363 uzrdgfni 12619 xptrrel 13567 rlimpm 14079 xpsc0 16043 xpsc1 16044 xpsfrnel2 16048 isohom 16259 ledm 17047 gsumxp 18198 dprd2d2 18266 tsmsxp 21768 dvbssntr 23470 esum2d 29482 poimirlem3 32582 rtrclex 36943 trclexi 36946 rtrclexi 36947 cnvtrcl0 36952 dmtrcl 36953 rp-imass 37085 rfovcnvf1od 37318 issmflem 39613 |
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