Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > concompss | Structured version Visualization version GIF version |
Description: The connected component containing 𝐴 is a superset of any other connected set containing 𝐴. (Contributed by Mario Carneiro, 19-Mar-2015.) |
Ref | Expression |
---|---|
concomp.2 | ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)} |
Ref | Expression |
---|---|
concompss | ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Con) → 𝑇 ⊆ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp1 1054 | . . . . 5 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Con) → 𝑇 ⊆ 𝑋) | |
2 | contop 21030 | . . . . . . 7 ⊢ ((𝐽 ↾t 𝑇) ∈ Con → (𝐽 ↾t 𝑇) ∈ Top) | |
3 | 2 | 3ad2ant3 1077 | . . . . . 6 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Con) → (𝐽 ↾t 𝑇) ∈ Top) |
4 | restrcl 20771 | . . . . . . 7 ⊢ ((𝐽 ↾t 𝑇) ∈ Top → (𝐽 ∈ V ∧ 𝑇 ∈ V)) | |
5 | 4 | simprd 478 | . . . . . 6 ⊢ ((𝐽 ↾t 𝑇) ∈ Top → 𝑇 ∈ V) |
6 | elpwg 4116 | . . . . . 6 ⊢ (𝑇 ∈ V → (𝑇 ∈ 𝒫 𝑋 ↔ 𝑇 ⊆ 𝑋)) | |
7 | 3, 5, 6 | 3syl 18 | . . . . 5 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Con) → (𝑇 ∈ 𝒫 𝑋 ↔ 𝑇 ⊆ 𝑋)) |
8 | 1, 7 | mpbird 246 | . . . 4 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Con) → 𝑇 ∈ 𝒫 𝑋) |
9 | 3simpc 1053 | . . . 4 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Con) → (𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Con)) | |
10 | eleq2 2677 | . . . . . 6 ⊢ (𝑦 = 𝑇 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑇)) | |
11 | oveq2 6557 | . . . . . . 7 ⊢ (𝑦 = 𝑇 → (𝐽 ↾t 𝑦) = (𝐽 ↾t 𝑇)) | |
12 | 11 | eleq1d 2672 | . . . . . 6 ⊢ (𝑦 = 𝑇 → ((𝐽 ↾t 𝑦) ∈ Con ↔ (𝐽 ↾t 𝑇) ∈ Con)) |
13 | 10, 12 | anbi12d 743 | . . . . 5 ⊢ (𝑦 = 𝑇 → ((𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Con) ↔ (𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Con))) |
14 | eleq2 2677 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)) | |
15 | oveq2 6557 | . . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝐽 ↾t 𝑥) = (𝐽 ↾t 𝑦)) | |
16 | 15 | eleq1d 2672 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝐽 ↾t 𝑥) ∈ Con ↔ (𝐽 ↾t 𝑦) ∈ Con)) |
17 | 14, 16 | anbi12d 743 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con) ↔ (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Con))) |
18 | 17 | cbvrabv 3172 | . . . . 5 ⊢ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)} = {𝑦 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑦 ∧ (𝐽 ↾t 𝑦) ∈ Con)} |
19 | 13, 18 | elrab2 3333 | . . . 4 ⊢ (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)} ↔ (𝑇 ∈ 𝒫 𝑋 ∧ (𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Con))) |
20 | 8, 9, 19 | sylanbrc 695 | . . 3 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Con) → 𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)}) |
21 | elssuni 4403 | . . 3 ⊢ (𝑇 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)} → 𝑇 ⊆ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)}) | |
22 | 20, 21 | syl 17 | . 2 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Con) → 𝑇 ⊆ ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)}) |
23 | concomp.2 | . 2 ⊢ 𝑆 = ∪ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴 ∈ 𝑥 ∧ (𝐽 ↾t 𝑥) ∈ Con)} | |
24 | 22, 23 | syl6sseqr 3615 | 1 ⊢ ((𝑇 ⊆ 𝑋 ∧ 𝐴 ∈ 𝑇 ∧ (𝐽 ↾t 𝑇) ∈ Con) → 𝑇 ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {crab 2900 Vcvv 3173 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 (class class class)co 6549 ↾t crest 15904 Topctop 20517 Conccon 21024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-rest 15906 df-top 20521 df-con 21025 |
This theorem is referenced by: concompcld 21047 tgpconcompeqg 21725 tgpconcomp 21726 |
Copyright terms: Public domain | W3C validator |