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Mirrors > Home > ILE Home > Th. List > cnfval | Unicode version |
Description: The set of all continuous functions from topology to topology . (Contributed by NM, 17-Oct-2006.) (Revised by Mario Carneiro, 21-Aug-2015.) |
Ref | Expression |
---|---|
cnfval | TopOn TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-cn 12360 | . . 3 | |
2 | 1 | a1i 9 | . 2 TopOn TopOn |
3 | simprr 521 | . . . . . 6 TopOn TopOn | |
4 | 3 | unieqd 3747 | . . . . 5 TopOn TopOn |
5 | toponuni 12185 | . . . . . 6 TopOn | |
6 | 5 | ad2antlr 480 | . . . . 5 TopOn TopOn |
7 | 4, 6 | eqtr4d 2175 | . . . 4 TopOn TopOn |
8 | simprl 520 | . . . . . 6 TopOn TopOn | |
9 | 8 | unieqd 3747 | . . . . 5 TopOn TopOn |
10 | toponuni 12185 | . . . . . 6 TopOn | |
11 | 10 | ad2antrr 479 | . . . . 5 TopOn TopOn |
12 | 9, 11 | eqtr4d 2175 | . . . 4 TopOn TopOn |
13 | 7, 12 | oveq12d 5792 | . . 3 TopOn TopOn |
14 | 8 | eleq2d 2209 | . . . 4 TopOn TopOn |
15 | 3, 14 | raleqbidv 2638 | . . 3 TopOn TopOn |
16 | 13, 15 | rabeqbidv 2681 | . 2 TopOn TopOn |
17 | topontop 12184 | . . 3 TopOn | |
18 | 17 | adantr 274 | . 2 TopOn TopOn |
19 | topontop 12184 | . . 3 TopOn | |
20 | 19 | adantl 275 | . 2 TopOn TopOn |
21 | fnmap 6549 | . . . 4 | |
22 | toponmax 12195 | . . . . . 6 TopOn | |
23 | 22 | elexd 2699 | . . . . 5 TopOn |
24 | 23 | adantl 275 | . . . 4 TopOn TopOn |
25 | toponmax 12195 | . . . . . 6 TopOn | |
26 | 25 | elexd 2699 | . . . . 5 TopOn |
27 | 26 | adantr 274 | . . . 4 TopOn TopOn |
28 | fnovex 5804 | . . . 4 | |
29 | 21, 24, 27, 28 | mp3an2i 1320 | . . 3 TopOn TopOn |
30 | rabexg 4071 | . . 3 | |
31 | 29, 30 | syl 14 | . 2 TopOn TopOn |
32 | 2, 16, 18, 20, 31 | ovmpod 5898 | 1 TopOn TopOn |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wceq 1331 wcel 1480 wral 2416 crab 2420 cvv 2686 cuni 3736 cxp 4537 ccnv 4538 cima 4542 wfn 5118 cfv 5123 (class class class)co 5774 cmpo 5776 cmap 6542 ctop 12167 TopOnctopon 12180 ccn 12357 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-top 12168 df-topon 12181 df-cn 12360 |
This theorem is referenced by: cnovex 12368 iscn 12369 |
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