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Mirrors > Home > MPE Home > Th. List > mreexdomd | Structured version Visualization version GIF version |
Description: In a Moore system whose closure operator has the exchange property, if 𝑆 is independent and contained in the closure of 𝑇, and either 𝑆 or 𝑇 is finite, then 𝑇 dominates 𝑆. This is an immediate consequence of mreexexd 17693. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mreexdomd.1 | ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
mreexdomd.2 | ⊢ 𝑁 = (mrCls‘𝐴) |
mreexdomd.3 | ⊢ 𝐼 = (mrInd‘𝐴) |
mreexdomd.4 | ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) |
mreexdomd.5 | ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) |
mreexdomd.6 | ⊢ (𝜑 → 𝑇 ⊆ 𝑋) |
mreexdomd.7 | ⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin)) |
mreexdomd.8 | ⊢ (𝜑 → 𝑆 ∈ 𝐼) |
Ref | Expression |
---|---|
mreexdomd | ⊢ (𝜑 → 𝑆 ≼ 𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mreexdomd.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | |
2 | mreexdomd.2 | . . 3 ⊢ 𝑁 = (mrCls‘𝐴) | |
3 | mreexdomd.3 | . . 3 ⊢ 𝐼 = (mrInd‘𝐴) | |
4 | mreexdomd.4 | . . 3 ⊢ (𝜑 → ∀𝑠 ∈ 𝒫 𝑋∀𝑦 ∈ 𝑋 ∀𝑧 ∈ ((𝑁‘(𝑠 ∪ {𝑦})) ∖ (𝑁‘𝑠))𝑦 ∈ (𝑁‘(𝑠 ∪ {𝑧}))) | |
5 | mreexdomd.8 | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ 𝐼) | |
6 | 3, 1, 5 | mrissd 17681 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
7 | dif0 4384 | . . . 4 ⊢ (𝑋 ∖ ∅) = 𝑋 | |
8 | 6, 7 | sseqtrrdi 4047 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑋 ∖ ∅)) |
9 | mreexdomd.6 | . . . 4 ⊢ (𝜑 → 𝑇 ⊆ 𝑋) | |
10 | 9, 7 | sseqtrrdi 4047 | . . 3 ⊢ (𝜑 → 𝑇 ⊆ (𝑋 ∖ ∅)) |
11 | mreexdomd.5 | . . . 4 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑇)) | |
12 | un0 4400 | . . . . 5 ⊢ (𝑇 ∪ ∅) = 𝑇 | |
13 | 12 | fveq2i 6910 | . . . 4 ⊢ (𝑁‘(𝑇 ∪ ∅)) = (𝑁‘𝑇) |
14 | 11, 13 | sseqtrrdi 4047 | . . 3 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘(𝑇 ∪ ∅))) |
15 | un0 4400 | . . . 4 ⊢ (𝑆 ∪ ∅) = 𝑆 | |
16 | 15, 5 | eqeltrid 2843 | . . 3 ⊢ (𝜑 → (𝑆 ∪ ∅) ∈ 𝐼) |
17 | mreexdomd.7 | . . 3 ⊢ (𝜑 → (𝑆 ∈ Fin ∨ 𝑇 ∈ Fin)) | |
18 | 1, 2, 3, 4, 8, 10, 14, 16, 17 | mreexexd 17693 | . 2 ⊢ (𝜑 → ∃𝑖 ∈ 𝒫 𝑇(𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼)) |
19 | simprrl 781 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑆 ≈ 𝑖) | |
20 | simprl 771 | . . . . 5 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ∈ 𝒫 𝑇) | |
21 | 20 | elpwid 4614 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ⊆ 𝑇) |
22 | 1 | elfvexd 6946 | . . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ V) |
23 | 22, 9 | ssexd 5330 | . . . . . 6 ⊢ (𝜑 → 𝑇 ∈ V) |
24 | ssdomg 9039 | . . . . . 6 ⊢ (𝑇 ∈ V → (𝑖 ⊆ 𝑇 → 𝑖 ≼ 𝑇)) | |
25 | 23, 24 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑖 ⊆ 𝑇 → 𝑖 ≼ 𝑇)) |
26 | 25 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → (𝑖 ⊆ 𝑇 → 𝑖 ≼ 𝑇)) |
27 | 21, 26 | mpd 15 | . . 3 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑖 ≼ 𝑇) |
28 | endomtr 9051 | . . 3 ⊢ ((𝑆 ≈ 𝑖 ∧ 𝑖 ≼ 𝑇) → 𝑆 ≼ 𝑇) | |
29 | 19, 27, 28 | syl2anc 584 | . 2 ⊢ ((𝜑 ∧ (𝑖 ∈ 𝒫 𝑇 ∧ (𝑆 ≈ 𝑖 ∧ (𝑖 ∪ ∅) ∈ 𝐼))) → 𝑆 ≼ 𝑇) |
30 | 18, 29 | rexlimddv 3159 | 1 ⊢ (𝜑 → 𝑆 ≼ 𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∀wral 3059 Vcvv 3478 ∖ cdif 3960 ∪ cun 3961 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 {csn 4631 class class class wbr 5148 ‘cfv 6563 ≈ cen 8981 ≼ cdom 8982 Fincfn 8984 Moorecmre 17627 mrClscmrc 17628 mrIndcmri 17629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-om 7888 df-1o 8505 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-mre 17631 df-mrc 17632 df-mri 17633 |
This theorem is referenced by: mreexfidimd 17695 |
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