![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > t1hmph | Structured version Visualization version GIF version |
Description: T1 is a topological property. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
t1hmph | ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Fre → 𝐾 ∈ Fre)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | t1top 21935 | . 2 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top) | |
2 | cnt1 21955 | . 2 ⊢ ((𝐽 ∈ Fre ∧ 𝑓:∪ 𝐾–1-1→∪ 𝐽 ∧ 𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ Fre) | |
3 | 1, 2 | haushmphlem 22392 | 1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Fre → 𝐾 ∈ Fre)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ∪ cuni 4800 class class class wbr 5030 Frect1 21912 ≃ 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-map 8391 df-top 21499 df-topon 21516 df-cld 21624 df-cn 21832 df-t1 21919 df-hmeo 22360 df-hmph 22361 |
This theorem is referenced by: t1r0 22426 ist1-5 22427 |
Copyright terms: Public domain | W3C validator |