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Mirrors > Home > MPE Home > Th. List > dmxp | Structured version Visualization version GIF version |
Description: The domain of a Cartesian product. Part of Theorem 3.13(x) of [Monk1] p. 37. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
dmxp | ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-xp 5561 | . . 3 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
2 | 1 | dmeqi 5773 | . 2 ⊢ dom (𝐴 × 𝐵) = dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
3 | n0 4310 | . . . . 5 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐵) | |
4 | 3 | biimpi 218 | . . . 4 ⊢ (𝐵 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐵) |
5 | 4 | ralrimivw 3183 | . . 3 ⊢ (𝐵 ≠ ∅ → ∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐵) |
6 | dmopab3 5788 | . . 3 ⊢ (∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐵 ↔ dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = 𝐴) | |
7 | 5, 6 | sylib 220 | . 2 ⊢ (𝐵 ≠ ∅ → dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} = 𝐴) |
8 | 2, 7 | syl5eq 2868 | 1 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∅c0 4291 {copab 5128 × cxp 5553 dom cdm 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-xp 5561 df-dm 5565 |
This theorem is referenced by: dmxpid 5800 rnxp 6027 dmxpss 6028 ssxpb 6031 relrelss 6124 unixp 6133 xpexr2 7624 xpexcnv 7625 frxp 7820 mpocurryd 7935 fodomr 8668 nqerf 10352 dmtrclfv 14378 pwsbas 16760 pwsle 16765 imasaddfnlem 16801 imasvscafn 16810 efgrcl 18841 frlmip 20922 txindislem 22241 metustexhalf 23166 rrxip 23993 dveq0 24597 dv11cn 24598 mbfmcst 31517 eulerpartlemt 31629 0rrv 31709 noxp1o 33170 noextendseq 33174 bdayfo 33182 noetalem3 33219 noetalem4 33220 curf 34885 curunc 34889 ismgmOLD 35143 diophrw 39376 |
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