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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 16742 and mrieqv2d 16743. 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 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝐴 ∈ (Moore‘𝑋)) |
3 | mrieqvlemd.2 | . . . 4 ⊢ 𝑁 = (mrCls‘𝐴) | |
4 | undif1 4344 | . . . . . 6 ⊢ ((𝑆 ∖ {𝑌}) ∪ {𝑌}) = (𝑆 ∪ {𝑌}) | |
5 | mrieqvlemd.3 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) | |
6 | 5 | adantr 481 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆 ⊆ 𝑋) |
7 | 6 | ssdifssd 4046 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑋) |
8 | 2, 3, 7 | mrcssidd 16729 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌}))) |
9 | simpr 485 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) | |
10 | 9 | snssd 4655 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → {𝑌} ⊆ (𝑁‘(𝑆 ∖ {𝑌}))) |
11 | 8, 10 | unssd 4089 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → ((𝑆 ∖ {𝑌}) ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌}))) |
12 | 4, 11 | eqsstrrid 3943 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌}))) |
13 | 12 | unssad 4090 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆 ⊆ (𝑁‘(𝑆 ∖ {𝑌}))) |
14 | difssd 4036 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑆) | |
15 | 2, 3, 13, 14 | mressmrcd 16731 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁‘𝑆) = (𝑁‘(𝑆 ∖ {𝑌}))) |
16 | 15 | eqcomd 2803 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) |
17 | 1, 3, 5 | mrcssidd 16729 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑆)) |
18 | mrieqvlemd.4 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑆) | |
19 | 17, 18 | sseldd 3896 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘𝑆)) |
20 | 19 | adantr 481 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) → 𝑌 ∈ (𝑁‘𝑆)) |
21 | simpr 485 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) | |
22 | 20, 21 | eleqtrrd 2888 | . 2 ⊢ ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) |
23 | 16, 22 | impbida 797 | 1 ⊢ (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1525 ∈ wcel 2083 ∖ cdif 3862 ∪ cun 3863 ⊆ wss 3865 {csn 4478 ‘cfv 6232 Moorecmre 16686 mrClscmrc 16687 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-int 4789 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-fv 6240 df-mre 16690 df-mrc 16691 |
This theorem is referenced by: mrieqvd 16742 mrieqv2d 16743 |
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