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Mirrors > Home > ILE Home > Th. List > cnpfval | Unicode version |
Description: The function mapping the points in a topology to the set of all functions from to topology continuous at that point. (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cnpfval | TopOn TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cnp 12285 | . . 3 | |
2 | 1 | a1i 9 | . 2 TopOn TopOn |
3 | simprl 505 | . . . . 5 TopOn TopOn | |
4 | 3 | unieqd 3717 | . . . 4 TopOn TopOn |
5 | toponuni 12109 | . . . . 5 TopOn | |
6 | 5 | ad2antrr 479 | . . . 4 TopOn TopOn |
7 | 4, 6 | eqtr4d 2153 | . . 3 TopOn TopOn |
8 | simprr 506 | . . . . . . 7 TopOn TopOn | |
9 | 8 | unieqd 3717 | . . . . . 6 TopOn TopOn |
10 | toponuni 12109 | . . . . . . 7 TopOn | |
11 | 10 | ad2antlr 480 | . . . . . 6 TopOn TopOn |
12 | 9, 11 | eqtr4d 2153 | . . . . 5 TopOn TopOn |
13 | 12, 7 | oveq12d 5760 | . . . 4 TopOn TopOn |
14 | 3 | rexeqdv 2610 | . . . . . 6 TopOn TopOn |
15 | 14 | imbi2d 229 | . . . . 5 TopOn TopOn |
16 | 8, 15 | raleqbidv 2615 | . . . 4 TopOn TopOn |
17 | 13, 16 | rabeqbidv 2655 | . . 3 TopOn TopOn |
18 | 7, 17 | mpteq12dv 3980 | . 2 TopOn TopOn |
19 | topontop 12108 | . . 3 TopOn | |
20 | 19 | adantr 274 | . 2 TopOn TopOn |
21 | topontop 12108 | . . 3 TopOn | |
22 | 21 | adantl 275 | . 2 TopOn TopOn |
23 | fnmap 6517 | . . . . . . . 8 | |
24 | 23 | a1i 9 | . . . . . . 7 TopOn TopOn |
25 | toponmax 12119 | . . . . . . . . 9 TopOn | |
26 | 25 | elexd 2673 | . . . . . . . 8 TopOn |
27 | 26 | adantl 275 | . . . . . . 7 TopOn TopOn |
28 | toponmax 12119 | . . . . . . . . 9 TopOn | |
29 | 28 | elexd 2673 | . . . . . . . 8 TopOn |
30 | 29 | adantr 274 | . . . . . . 7 TopOn TopOn |
31 | fnovex 5772 | . . . . . . 7 | |
32 | 24, 27, 30, 31 | syl3anc 1201 | . . . . . 6 TopOn TopOn |
33 | 32 | adantr 274 | . . . . 5 TopOn TopOn |
34 | ssrab2 3152 | . . . . . 6 | |
35 | elpw2g 4051 | . . . . . 6 | |
36 | 34, 35 | mpbiri 167 | . . . . 5 |
37 | 33, 36 | syl 14 | . . . 4 TopOn TopOn |
38 | 37 | fmpttd 5543 | . . 3 TopOn TopOn |
39 | 28 | adantr 274 | . . 3 TopOn TopOn |
40 | 32 | pwexd 4075 | . . 3 TopOn TopOn |
41 | fex2 5261 | . . 3 | |
42 | 38, 39, 40, 41 | syl3anc 1201 | . 2 TopOn TopOn |
43 | 2, 18, 20, 22, 42 | ovmpod 5866 | 1 TopOn TopOn |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1316 wcel 1465 wral 2393 wrex 2394 crab 2397 cvv 2660 wss 3041 cpw 3480 cuni 3706 cmpt 3959 cxp 4507 cima 4512 wfn 5088 wf 5089 cfv 5093 (class class class)co 5742 cmpo 5744 cmap 6510 ctop 12091 TopOnctopon 12104 ccnp 12282 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-ral 2398 df-rex 2399 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-fv 5101 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-map 6512 df-top 12092 df-topon 12105 df-cnp 12285 |
This theorem is referenced by: cnpval 12294 |
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