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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 23764 | . . 3 ⊢ ≃ = (◡Homeo “ (V ∖ 1o)) | |
| 2 | cnvimass 6100 | . . . 4 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ dom Homeo | |
| 3 | hmeofn 23765 | . . . . 5 ⊢ Homeo Fn (Top × Top) | |
| 4 | fndm 6671 | . . . . 5 ⊢ (Homeo Fn (Top × Top) → dom Homeo = (Top × Top)) | |
| 5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ dom Homeo = (Top × Top) | 
| 6 | 2, 5 | sseqtri 4032 | . . 3 ⊢ (◡Homeo “ (V ∖ 1o)) ⊆ (Top × Top) | 
| 7 | 1, 6 | eqsstri 4030 | . 2 ⊢ ≃ ⊆ (Top × Top) | 
| 8 | 7 | brel 5750 | 1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 class class class wbr 5143 × cxp 5683 ◡ccnv 5684 dom cdm 5685 “ cima 5688 Fn wfn 6556 1oc1o 8499 Topctop 22899 Homeochmeo 23761 ≃ chmph 23762 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-hmeo 23763 df-hmph 23764 | 
| This theorem is referenced by: hmphtop1 23787 hmphtop2 23788 | 
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