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Mirrors > Home > MPE Home > Th. List > hmphref | Structured version Visualization version GIF version |
Description: "Is homeomorphic to" is reflexive. (Contributed by FL, 25-Feb-2007.) (Proof shortened by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
hmphref | ⊢ (𝐽 ∈ Top → 𝐽 ≃ 𝐽) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | toptopon2 21242 | . . 3 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽)) | |
2 | idhmeo 22097 | . . 3 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → ( I ↾ ∪ 𝐽) ∈ (𝐽Homeo𝐽)) | |
3 | 1, 2 | sylbi 209 | . 2 ⊢ (𝐽 ∈ Top → ( I ↾ ∪ 𝐽) ∈ (𝐽Homeo𝐽)) |
4 | hmphi 22101 | . 2 ⊢ (( I ↾ ∪ 𝐽) ∈ (𝐽Homeo𝐽) → 𝐽 ≃ 𝐽) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝐽 ∈ Top → 𝐽 ≃ 𝐽) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2050 ∪ cuni 4708 class class class wbr 4925 I cid 5307 ↾ cres 5405 ‘cfv 6185 (class class class)co 6974 Topctop 21217 TopOnctopon 21234 Homeochmeo 22077 ≃ chmph 22078 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-suc 6032 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-ov 6977 df-oprab 6978 df-mpo 6979 df-1st 7499 df-2nd 7500 df-1o 7903 df-map 8206 df-top 21218 df-topon 21235 df-cn 21551 df-hmeo 22079 df-hmph 22080 |
This theorem is referenced by: hmpher 22108 hmph0 22119 |
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