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Mirrors > Home > MPE Home > Th. List > haushmphlem | Structured version Visualization version Unicode version |
Description: Lemma for haushmph 21595 and similar theorems. If the topological property is preserved under injective preimages, then property is preserved under homeomorphisms. (Contributed by Mario Carneiro, 25-Aug-2015.) |
Ref | Expression |
---|---|
haushmphlem.1 | |
haushmphlem.2 |
Ref | Expression |
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haushmphlem |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hmphsym 21585 | . 2 | |
2 | hmph 21579 | . . 3 | |
3 | n0 3931 | . . . 4 | |
4 | simpl 473 | . . . . . . 7 | |
5 | eqid 2622 | . . . . . . . . . 10 | |
6 | eqid 2622 | . . . . . . . . . 10 | |
7 | 5, 6 | hmeof1o 21567 | . . . . . . . . 9 |
8 | 7 | adantl 482 | . . . . . . . 8 |
9 | f1of1 6136 | . . . . . . . 8 | |
10 | 8, 9 | syl 17 | . . . . . . 7 |
11 | hmeocn 21563 | . . . . . . . 8 | |
12 | 11 | adantl 482 | . . . . . . 7 |
13 | haushmphlem.2 | . . . . . . 7 | |
14 | 4, 10, 12, 13 | syl3anc 1326 | . . . . . 6 |
15 | 14 | expcom 451 | . . . . 5 |
16 | 15 | exlimiv 1858 | . . . 4 |
17 | 3, 16 | sylbi 207 | . . 3 |
18 | 2, 17 | sylbi 207 | . 2 |
19 | 1, 18 | syl 17 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 w3a 1037 wex 1704 wcel 1990 wne 2794 c0 3915 cuni 4436 class class class wbr 4653 wf1 5885 wf1o 5887 (class class class)co 6650 ctop 20698 ccn 21028 chmeo 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-pw 4160 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-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-1o 7560 df-map 7859 df-top 20699 df-topon 20716 df-cn 21031 df-hmeo 21558 df-hmph 21559 |
This theorem is referenced by: t0hmph 21593 t1hmph 21594 haushmph 21595 |
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