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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 5563 | . . 3 ⊢ (𝐴 × 𝐵) = {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} | |
2 | 1 | dmeqi 5775 | . 2 ⊢ dom (𝐴 × 𝐵) = dom {〈𝑦, 𝑥〉 ∣ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵)} |
3 | n0 4312 | . . . . 5 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐵) | |
4 | 3 | biimpi 218 | . . . 4 ⊢ (𝐵 ≠ ∅ → ∃𝑥 𝑥 ∈ 𝐵) |
5 | 4 | ralrimivw 3185 | . . 3 ⊢ (𝐵 ≠ ∅ → ∀𝑦 ∈ 𝐴 ∃𝑥 𝑥 ∈ 𝐵) |
6 | dmopab3 5790 | . . 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 2870 | 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 3018 ∀wral 3140 ∅c0 4293 {copab 5130 × cxp 5555 dom cdm 5557 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-xp 5563 df-dm 5567 |
This theorem is referenced by: dmxpid 5802 rnxp 6029 dmxpss 6030 ssxpb 6033 relrelss 6126 unixp 6135 xpexr2 7626 xpexcnv 7627 frxp 7822 mpocurryd 7937 fodomr 8670 nqerf 10354 dmtrclfv 14380 pwsbas 16762 pwsle 16767 imasaddfnlem 16803 imasvscafn 16812 efgrcl 18843 frlmip 20924 txindislem 22243 metustexhalf 23168 rrxip 23995 dveq0 24599 dv11cn 24600 mbfmcst 31519 eulerpartlemt 31631 0rrv 31711 noxp1o 33172 noextendseq 33176 bdayfo 33184 noetalem3 33221 noetalem4 33222 curf 34872 curunc 34876 ismgmOLD 35130 diophrw 39363 |
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