![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mrieqvlemd | Structured version Visualization version GIF version |
Description: In a Moore system, if 𝑌 is a member of 𝑆, (𝑆 ∖ {𝑌}) and 𝑆 have the same closure if and only if 𝑌 is in the closure of (𝑆 ∖ {𝑌}). Used in the proof of mrieqvd 17581 and mrieqv2d 17582. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
mrieqvlemd.1 | ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) |
mrieqvlemd.2 | ⊢ 𝑁 = (mrCls‘𝐴) |
mrieqvlemd.3 | ⊢ (𝜑 → 𝑆 ⊆ 𝑋) |
mrieqvlemd.4 | ⊢ (𝜑 → 𝑌 ∈ 𝑆) |
Ref | Expression |
---|---|
mrieqvlemd | ⊢ (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrieqvlemd.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ (Moore‘𝑋)) | |
2 | 1 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝐴 ∈ (Moore‘𝑋)) |
3 | mrieqvlemd.2 | . . . 4 ⊢ 𝑁 = (mrCls‘𝐴) | |
4 | undif1 4467 | . . . . . 6 ⊢ ((𝑆 ∖ {𝑌}) ∪ {𝑌}) = (𝑆 ∪ {𝑌}) | |
5 | mrieqvlemd.3 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) | |
6 | 5 | adantr 480 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆 ⊆ 𝑋) |
7 | 6 | ssdifssd 4134 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑋) |
8 | 2, 3, 7 | mrcssidd 17568 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌}))) |
9 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) | |
10 | 9 | snssd 4804 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → {𝑌} ⊆ (𝑁‘(𝑆 ∖ {𝑌}))) |
11 | 8, 10 | unssd 4178 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → ((𝑆 ∖ {𝑌}) ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌}))) |
12 | 4, 11 | eqsstrrid 4023 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌}))) |
13 | 12 | unssad 4179 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆 ⊆ (𝑁‘(𝑆 ∖ {𝑌}))) |
14 | difssd 4124 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑆) | |
15 | 2, 3, 13, 14 | mressmrcd 17570 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁‘𝑆) = (𝑁‘(𝑆 ∖ {𝑌}))) |
16 | 15 | eqcomd 2730 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) |
17 | 1, 3, 5 | mrcssidd 17568 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑆)) |
18 | mrieqvlemd.4 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑆) | |
19 | 17, 18 | sseldd 3975 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘𝑆)) |
20 | 19 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) → 𝑌 ∈ (𝑁‘𝑆)) |
21 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) | |
22 | 20, 21 | eleqtrrd 2828 | . 2 ⊢ ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) |
23 | 16, 22 | impbida 798 | 1 ⊢ (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∖ cdif 3937 ∪ cun 3938 ⊆ wss 3940 {csn 4620 ‘cfv 6533 Moorecmre 17525 mrClscmrc 17526 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fv 6541 df-mre 17529 df-mrc 17530 |
This theorem is referenced by: mrieqvd 17581 mrieqv2d 17582 |
Copyright terms: Public domain | W3C validator |