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Mirrors > Home > MPE Home > Th. List > hmpher | Structured version Visualization version GIF version |
Description: "Is homeomorphic to" is an equivalence relation. (Contributed by FL, 22-Mar-2007.) (Revised by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
hmpher | ⊢ ≃ Er Top |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-hmph 22361 | . . . 4 ⊢ ≃ = (◡Homeo “ (V ∖ 1o)) | |
2 | cnvimass 5916 | . . . . 5 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ dom Homeo | |
3 | hmeofn 22362 | . . . . . 6 ⊢ Homeo Fn (Top × Top) | |
4 | 3 | fndmi 6426 | . . . . 5 ⊢ dom Homeo = (Top × Top) |
5 | 2, 4 | sseqtri 3951 | . . . 4 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ (Top × Top) |
6 | 1, 5 | eqsstri 3949 | . . 3 ⊢ ≃ ⊆ (Top × Top) |
7 | relxp 5537 | . . 3 ⊢ Rel (Top × Top) | |
8 | relss 5620 | . . 3 ⊢ ( ≃ ⊆ (Top × Top) → (Rel (Top × Top) → Rel ≃ )) | |
9 | 6, 7, 8 | mp2 9 | . 2 ⊢ Rel ≃ |
10 | hmphsym 22387 | . 2 ⊢ (𝑥 ≃ 𝑦 → 𝑦 ≃ 𝑥) | |
11 | hmphtr 22388 | . 2 ⊢ ((𝑥 ≃ 𝑦 ∧ 𝑦 ≃ 𝑧) → 𝑥 ≃ 𝑧) | |
12 | hmphref 22386 | . . 3 ⊢ (𝑥 ∈ Top → 𝑥 ≃ 𝑥) | |
13 | hmphtop1 22384 | . . 3 ⊢ (𝑥 ≃ 𝑥 → 𝑥 ∈ Top) | |
14 | 12, 13 | impbii 212 | . 2 ⊢ (𝑥 ∈ Top ↔ 𝑥 ≃ 𝑥) |
15 | 9, 10, 11, 14 | iseri 8299 | 1 ⊢ ≃ Er Top |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 class class class wbr 5030 × cxp 5517 ◡ccnv 5518 dom cdm 5519 “ cima 5522 Rel wrel 5524 1oc1o 8078 Er wer 8269 Topctop 21498 Homeochmeo 22358 ≃ chmph 22359 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-1o 8085 df-er 8272 df-map 8391 df-top 21499 df-topon 21516 df-cn 21832 df-hmeo 22360 df-hmph 22361 |
This theorem is referenced by: ismntop 31377 |
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