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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 21559 | . . 3 ⊢ ≃ = (◡Homeo “ (V ∖ 1𝑜)) | |
2 | cnvimass 5485 | . . . 4 ⊢ (◡Homeo “ (V ∖ 1𝑜)) ⊆ dom Homeo | |
3 | hmeofn 21560 | . . . . 5 ⊢ Homeo Fn (Top × Top) | |
4 | fndm 5990 | . . . . 5 ⊢ (Homeo Fn (Top × Top) → dom Homeo = (Top × Top)) | |
5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ dom Homeo = (Top × Top) |
6 | 2, 5 | sseqtri 3637 | . . 3 ⊢ (◡Homeo “ (V ∖ 1𝑜)) ⊆ (Top × Top) |
7 | 1, 6 | eqsstri 3635 | . 2 ⊢ ≃ ⊆ (Top × Top) |
8 | 7 | brel 5168 | 1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ Top ∧ 𝐾 ∈ Top)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 class class class wbr 4653 × cxp 5112 ◡ccnv 5113 dom cdm 5114 “ cima 5117 Fn wfn 5883 1𝑜c1o 7553 Topctop 20698 Homeochmeo 21556 ≃ chmph 21557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-hmeo 21558 df-hmph 21559 |
This theorem is referenced by: hmphtop1 21582 hmphtop2 21583 |
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