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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 23227 | . 2 ⊢ (𝐽 ∈ Fre → 𝐽 ∈ Top) | |
2 | cnt1 23247 | . 2 ⊢ ((𝐽 ∈ Fre ∧ 𝑓:∪ 𝐾–1-1→∪ 𝐽 ∧ 𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ Fre) | |
3 | 1, 2 | haushmphlem 23684 | 1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Fre → 𝐾 ∈ Fre)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2099 ∪ cuni 4903 class class class wbr 5142 Frect1 23204 ≃ chmph 23651 |
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 2699 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-2nd 7988 df-1o 8480 df-map 8840 df-top 22789 df-topon 22806 df-cld 22916 df-cn 23124 df-t1 23211 df-hmeo 23652 df-hmph 23653 |
This theorem is referenced by: t1r0 23718 ist1-5 23719 |
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