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Mirrors > Home > MPE Home > Th. List > hmphtop | Structured version Visualization version GIF version |
Description: Reverse closure for the homeomorphic predicate. (Contributed by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
hmphtop | ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-hmph 22052 | . . 3 ⊢ ≃ = (◡Homeo “ (V ∖ 1o)) | |
2 | cnvimass 5832 | . . . 4 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ dom Homeo | |
3 | hmeofn 22053 | . . . . 5 ⊢ Homeo Fn (Top × Top) | |
4 | fndm 6332 | . . . . 5 ⊢ (Homeo Fn (Top × Top) → dom Homeo = (Top × Top)) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ dom Homeo = (Top × Top) |
6 | 2, 5 | sseqtri 3930 | . . 3 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ (Top × Top) |
7 | 1, 6 | eqsstri 3928 | . 2 ⊢ ≃ ⊆ (Top × Top) |
8 | 7 | brel 5510 | 1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1525 ∈ wcel 2083 Vcvv 3440 ∖ cdif 3862 class class class wbr 4968 × cxp 5448 ◡ccnv 5449 dom cdm 5450 “ cima 5453 Fn wfn 6227 1oc1o 7953 Topctop 21189 Homeochmeo 22049 ≃ chmph 22050 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-rab 3116 df-v 3442 df-sbc 3712 df-csb 3818 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-nul 4218 df-if 4388 df-sn 4479 df-pr 4481 df-op 4485 df-uni 4752 df-iun 4833 df-br 4969 df-opab 5031 df-mpt 5048 df-id 5355 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-fv 6240 df-ov 7026 df-oprab 7027 df-mpo 7028 df-1st 7552 df-2nd 7553 df-hmeo 22051 df-hmph 22052 |
This theorem is referenced by: hmphtop1 22075 hmphtop2 22076 |
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