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Mirrors > Home > MPE Home > Th. List > tgcnp | Structured version Visualization version Unicode version |
Description: The "continuous at a point" predicate when the range is given by a basis for a topology. (Contributed by Mario Carneiro, 3-Feb-2015.) (Revised by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
tgcn.1 | TopOn |
tgcn.3 | |
tgcn.4 | TopOn |
tgcnp.5 |
Ref | Expression |
---|---|
tgcnp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tgcn.1 | . . . 4 TopOn | |
2 | tgcn.4 | . . . 4 TopOn | |
3 | tgcnp.5 | . . . 4 | |
4 | iscnp 21041 | . . . 4 TopOn TopOn | |
5 | 1, 2, 3, 4 | syl3anc 1326 | . . 3 |
6 | tgcn.3 | . . . . . . . . 9 | |
7 | topontop 20718 | . . . . . . . . . 10 TopOn | |
8 | 2, 7 | syl 17 | . . . . . . . . 9 |
9 | 6, 8 | eqeltrrd 2702 | . . . . . . . 8 |
10 | tgclb 20774 | . . . . . . . 8 | |
11 | 9, 10 | sylibr 224 | . . . . . . 7 |
12 | bastg 20770 | . . . . . . 7 | |
13 | 11, 12 | syl 17 | . . . . . 6 |
14 | 13, 6 | sseqtr4d 3642 | . . . . 5 |
15 | ssralv 3666 | . . . . 5 | |
16 | 14, 15 | syl 17 | . . . 4 |
17 | 16 | anim2d 589 | . . 3 |
18 | 5, 17 | sylbid 230 | . 2 |
19 | 6 | eleq2d 2687 | . . . . . . 7 |
20 | 19 | biimpa 501 | . . . . . 6 |
21 | tg2 20769 | . . . . . . . . 9 | |
22 | r19.29 3072 | . . . . . . . . . . 11 | |
23 | sstr 3611 | . . . . . . . . . . . . . . . . . 18 | |
24 | 23 | expcom 451 | . . . . . . . . . . . . . . . . 17 |
25 | 24 | anim2d 589 | . . . . . . . . . . . . . . . 16 |
26 | 25 | reximdv 3016 | . . . . . . . . . . . . . . 15 |
27 | 26 | com12 32 | . . . . . . . . . . . . . 14 |
28 | 27 | imim2i 16 | . . . . . . . . . . . . 13 |
29 | 28 | imp32 449 | . . . . . . . . . . . 12 |
30 | 29 | rexlimivw 3029 | . . . . . . . . . . 11 |
31 | 22, 30 | syl 17 | . . . . . . . . . 10 |
32 | 31 | expcom 451 | . . . . . . . . 9 |
33 | 21, 32 | syl 17 | . . . . . . . 8 |
34 | 33 | ex 450 | . . . . . . 7 |
35 | 34 | com23 86 | . . . . . 6 |
36 | 20, 35 | syl 17 | . . . . 5 |
37 | 36 | ralrimdva 2969 | . . . 4 |
38 | 37 | anim2d 589 | . . 3 |
39 | iscnp 21041 | . . . 4 TopOn TopOn | |
40 | 1, 2, 3, 39 | syl3anc 1326 | . . 3 |
41 | 38, 40 | sylibrd 249 | . 2 |
42 | 18, 41 | impbid 202 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wral 2912 wrex 2913 wss 3574 cima 5117 wf 5884 cfv 5888 (class class class)co 6650 ctg 16098 ctop 20698 TopOnctopon 20715 ctb 20749 ccnp 21029 |
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-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-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-map 7859 df-topgen 16104 df-top 20699 df-topon 20716 df-bases 20750 df-cnp 21032 |
This theorem is referenced by: txcnp 21423 ptcnp 21425 metcnp3 22345 |
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