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 16909 and mrieqv2d 16910. 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 483 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝐴 ∈ (Moore‘𝑋)) |
3 | mrieqvlemd.2 | . . . 4 ⊢ 𝑁 = (mrCls‘𝐴) | |
4 | undif1 4424 | . . . . . 6 ⊢ ((𝑆 ∖ {𝑌}) ∪ {𝑌}) = (𝑆 ∪ {𝑌}) | |
5 | mrieqvlemd.3 | . . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ 𝑋) | |
6 | 5 | adantr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆 ⊆ 𝑋) |
7 | 6 | ssdifssd 4119 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑋) |
8 | 2, 3, 7 | mrcssidd 16896 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌}))) |
9 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) | |
10 | 9 | snssd 4742 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → {𝑌} ⊆ (𝑁‘(𝑆 ∖ {𝑌}))) |
11 | 8, 10 | unssd 4162 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → ((𝑆 ∖ {𝑌}) ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌}))) |
12 | 4, 11 | eqsstrrid 4016 | . . . . 5 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∪ {𝑌}) ⊆ (𝑁‘(𝑆 ∖ {𝑌}))) |
13 | 12 | unssad 4163 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → 𝑆 ⊆ (𝑁‘(𝑆 ∖ {𝑌}))) |
14 | difssd 4109 | . . . 4 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑆 ∖ {𝑌}) ⊆ 𝑆) | |
15 | 2, 3, 13, 14 | mressmrcd 16898 | . . 3 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁‘𝑆) = (𝑁‘(𝑆 ∖ {𝑌}))) |
16 | 15 | eqcomd 2827 | . 2 ⊢ ((𝜑 ∧ 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) |
17 | 1, 3, 5 | mrcssidd 16896 | . . . . 5 ⊢ (𝜑 → 𝑆 ⊆ (𝑁‘𝑆)) |
18 | mrieqvlemd.4 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝑆) | |
19 | 17, 18 | sseldd 3968 | . . . 4 ⊢ (𝜑 → 𝑌 ∈ (𝑁‘𝑆)) |
20 | 19 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) → 𝑌 ∈ (𝑁‘𝑆)) |
21 | simpr 487 | . . 3 ⊢ ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) → (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) | |
22 | 20, 21 | eleqtrrd 2916 | . 2 ⊢ ((𝜑 ∧ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆)) → 𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌}))) |
23 | 16, 22 | impbida 799 | 1 ⊢ (𝜑 → (𝑌 ∈ (𝑁‘(𝑆 ∖ {𝑌})) ↔ (𝑁‘(𝑆 ∖ {𝑌})) = (𝑁‘𝑆))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∖ cdif 3933 ∪ cun 3934 ⊆ wss 3936 {csn 4567 ‘cfv 6355 Moorecmre 16853 mrClscmrc 16854 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fv 6363 df-mre 16857 df-mrc 16858 |
This theorem is referenced by: mrieqvd 16909 mrieqv2d 16910 |
Copyright terms: Public domain | W3C validator |