Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > prsssdm | Structured version Visualization version GIF version |
Description: Domain of a subproset relation. (Contributed by Thierry Arnoux, 12-Sep-2018.) |
Ref | Expression |
---|---|
ordtNEW.b | ⊢ 𝐵 = (Base‘𝐾) |
ordtNEW.l | ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) |
Ref | Expression |
---|---|
prsssdm | ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordtNEW.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | ordtNEW.l | . . . 4 ⊢ ≤ = ((le‘𝐾) ∩ (𝐵 × 𝐵)) | |
3 | 1, 2 | prsss 31438 | . . 3 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ( ≤ ∩ (𝐴 × 𝐴)) = ((le‘𝐾) ∩ (𝐴 × 𝐴))) |
4 | 3 | dmeqd 5748 | . 2 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = dom ((le‘𝐾) ∩ (𝐴 × 𝐴))) |
5 | 1 | ressprs 30815 | . . . 4 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐾 ↾s 𝐴) ∈ Proset ) |
6 | eqid 2738 | . . . . 5 ⊢ (Base‘(𝐾 ↾s 𝐴)) = (Base‘(𝐾 ↾s 𝐴)) | |
7 | eqid 2738 | . . . . 5 ⊢ ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) = ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) | |
8 | 6, 7 | prsdm 31436 | . . . 4 ⊢ ((𝐾 ↾s 𝐴) ∈ Proset → dom ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) = (Base‘(𝐾 ↾s 𝐴))) |
9 | 5, 8 | syl 17 | . . 3 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) = (Base‘(𝐾 ↾s 𝐴))) |
10 | eqid 2738 | . . . . . . . . 9 ⊢ (𝐾 ↾s 𝐴) = (𝐾 ↾s 𝐴) | |
11 | 10, 1 | ressbas2 16658 | . . . . . . . 8 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 = (Base‘(𝐾 ↾s 𝐴))) |
12 | fvex 6687 | . . . . . . . 8 ⊢ (Base‘(𝐾 ↾s 𝐴)) ∈ V | |
13 | 11, 12 | eqeltrdi 2841 | . . . . . . 7 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ∈ V) |
14 | eqid 2738 | . . . . . . . 8 ⊢ (le‘𝐾) = (le‘𝐾) | |
15 | 10, 14 | ressle 16775 | . . . . . . 7 ⊢ (𝐴 ∈ V → (le‘𝐾) = (le‘(𝐾 ↾s 𝐴))) |
16 | 13, 15 | syl 17 | . . . . . 6 ⊢ (𝐴 ⊆ 𝐵 → (le‘𝐾) = (le‘(𝐾 ↾s 𝐴))) |
17 | 16 | adantl 485 | . . . . 5 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (le‘𝐾) = (le‘(𝐾 ↾s 𝐴))) |
18 | 11 | adantl 485 | . . . . . 6 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → 𝐴 = (Base‘(𝐾 ↾s 𝐴))) |
19 | 18 | sqxpeqd 5557 | . . . . 5 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → (𝐴 × 𝐴) = ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴)))) |
20 | 17, 19 | ineq12d 4104 | . . . 4 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → ((le‘𝐾) ∩ (𝐴 × 𝐴)) = ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))) |
21 | 20 | dmeqd 5748 | . . 3 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = dom ((le‘(𝐾 ↾s 𝐴)) ∩ ((Base‘(𝐾 ↾s 𝐴)) × (Base‘(𝐾 ↾s 𝐴))))) |
22 | 9, 21, 18 | 3eqtr4d 2783 | . 2 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ((le‘𝐾) ∩ (𝐴 × 𝐴)) = 𝐴) |
23 | 4, 22 | eqtrd 2773 | 1 ⊢ ((𝐾 ∈ Proset ∧ 𝐴 ⊆ 𝐵) → dom ( ≤ ∩ (𝐴 × 𝐴)) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 Vcvv 3398 ∩ cin 3842 ⊆ wss 3843 × cxp 5523 dom cdm 5525 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 ↾s cress 16587 lecple 16675 Proset cproset 17652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-dec 12180 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-ple 16688 df-proset 17654 |
This theorem is referenced by: ordtrest2NEWlem 31444 ordtrest2NEW 31445 |
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