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Mirrors > Home > ILE Home > Th. List > tx1cn | Unicode version |
Description: Continuity of the first projection map of a topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
tx1cn | TopOn TopOn |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1stres 6057 | . . 3 | |
2 | 1 | a1i 9 | . 2 TopOn TopOn |
3 | toponss 12193 | . . . . . . . . . 10 TopOn | |
4 | 3 | adantlr 468 | . . . . . . . . 9 TopOn TopOn |
5 | xpss1 4649 | . . . . . . . . 9 | |
6 | 4, 5 | syl 14 | . . . . . . . 8 TopOn TopOn |
7 | 6 | sseld 3096 | . . . . . . 7 TopOn TopOn |
8 | 7 | pm4.71rd 391 | . . . . . 6 TopOn TopOn |
9 | ffn 5272 | . . . . . . . 8 | |
10 | elpreima 5539 | . . . . . . . 8 | |
11 | 1, 9, 10 | mp2b 8 | . . . . . . 7 |
12 | fvres 5445 | . . . . . . . . . 10 | |
13 | 12 | eleq1d 2208 | . . . . . . . . 9 |
14 | 1st2nd2 6073 | . . . . . . . . . 10 | |
15 | xp2nd 6064 | . . . . . . . . . 10 | |
16 | elxp6 6067 | . . . . . . . . . . . 12 | |
17 | anass 398 | . . . . . . . . . . . 12 | |
18 | an32 551 | . . . . . . . . . . . 12 | |
19 | 16, 17, 18 | 3bitr2i 207 | . . . . . . . . . . 11 |
20 | 19 | baib 904 | . . . . . . . . . 10 |
21 | 14, 15, 20 | syl2anc 408 | . . . . . . . . 9 |
22 | 13, 21 | bitr4d 190 | . . . . . . . 8 |
23 | 22 | pm5.32i 449 | . . . . . . 7 |
24 | 11, 23 | bitri 183 | . . . . . 6 |
25 | 8, 24 | syl6rbbr 198 | . . . . 5 TopOn TopOn |
26 | 25 | eqrdv 2137 | . . . 4 TopOn TopOn |
27 | toponmax 12192 | . . . . . 6 TopOn | |
28 | 27 | ad2antlr 480 | . . . . 5 TopOn TopOn |
29 | txopn 12434 | . . . . . 6 TopOn TopOn | |
30 | 29 | anassrs 397 | . . . . 5 TopOn TopOn |
31 | 28, 30 | mpdan 417 | . . . 4 TopOn TopOn |
32 | 26, 31 | eqeltrd 2216 | . . 3 TopOn TopOn |
33 | 32 | ralrimiva 2505 | . 2 TopOn TopOn |
34 | txtopon 12431 | . . 3 TopOn TopOn TopOn | |
35 | simpl 108 | . . 3 TopOn TopOn TopOn | |
36 | iscn 12366 | . . 3 TopOn TopOn | |
37 | 34, 35, 36 | syl2anc 408 | . 2 TopOn TopOn |
38 | 2, 33, 37 | mpbir2and 928 | 1 TopOn TopOn |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wb 104 wceq 1331 wcel 1480 wral 2416 wss 3071 cop 3530 cxp 4537 ccnv 4538 cres 4541 cima 4542 wfn 5118 wf 5119 cfv 5123 (class class class)co 5774 c1st 6036 c2nd 6037 TopOnctopon 12177 ccn 12354 ctx 12421 |
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-coll 4043 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-reu 2423 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-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-map 6544 df-topgen 12141 df-top 12165 df-topon 12178 df-bases 12210 df-cn 12357 df-tx 12422 |
This theorem is referenced by: txcn 12444 cnmpt1st 12457 |
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