![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > hmphi | Structured version Visualization version GIF version |
Description: If there is a homeomorphism between spaces, then the spaces are homeomorphic. (Contributed by Mario Carneiro, 23-Aug-2015.) |
Ref | Expression |
---|---|
hmphi | ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐽 ≃ 𝐾) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ne0i 4337 | . 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → (𝐽Homeo𝐾) ≠ ∅) | |
2 | hmph 23771 | . 2 ⊢ (𝐽 ≃ 𝐾 ↔ (𝐽Homeo𝐾) ≠ ∅) | |
3 | 1, 2 | sylibr 233 | 1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐽 ≃ 𝐾) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ≠ wne 2930 ∅c0 4325 class class class wbr 5153 (class class class)co 7424 Homeochmeo 23748 ≃ chmph 23749 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 ax-un 7746 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-suc 6382 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-fv 6562 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-1o 8496 df-hmeo 23750 df-hmph 23751 |
This theorem is referenced by: hmphref 23776 hmphsym 23777 hmphtr 23778 indishmph 23793 ptcmpfi 23808 t0kq 23813 kqhmph 23814 xrhmph 24963 xrge0hmph 33747 reheibor 37540 |
Copyright terms: Public domain | W3C validator |