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Mirrors > Home > ILE Home > Th. List > txuni2 | Unicode version |
Description: The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
txval.1 | |
txuni2.1 | |
txuni2.2 |
Ref | Expression |
---|---|
txuni2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relxp 4648 | . . 3 | |
2 | txuni2.1 | . . . . . . . 8 | |
3 | 2 | eleq2i 2206 | . . . . . . 7 |
4 | eluni2 3740 | . . . . . . 7 | |
5 | 3, 4 | bitri 183 | . . . . . 6 |
6 | txuni2.2 | . . . . . . . 8 | |
7 | 6 | eleq2i 2206 | . . . . . . 7 |
8 | eluni2 3740 | . . . . . . 7 | |
9 | 7, 8 | bitri 183 | . . . . . 6 |
10 | 5, 9 | anbi12i 455 | . . . . 5 |
11 | opelxp 4569 | . . . . 5 | |
12 | reeanv 2600 | . . . . 5 | |
13 | 10, 11, 12 | 3bitr4i 211 | . . . 4 |
14 | opelxp 4569 | . . . . . 6 | |
15 | eqid 2139 | . . . . . . . . . 10 | |
16 | xpeq1 4553 | . . . . . . . . . . . 12 | |
17 | 16 | eqeq2d 2151 | . . . . . . . . . . 11 |
18 | xpeq2 4554 | . . . . . . . . . . . 12 | |
19 | 18 | eqeq2d 2151 | . . . . . . . . . . 11 |
20 | 17, 19 | rspc2ev 2804 | . . . . . . . . . 10 |
21 | 15, 20 | mp3an3 1304 | . . . . . . . . 9 |
22 | vex 2689 | . . . . . . . . . . 11 | |
23 | vex 2689 | . . . . . . . . . . 11 | |
24 | 22, 23 | xpex 4654 | . . . . . . . . . 10 |
25 | eqeq1 2146 | . . . . . . . . . . 11 | |
26 | 25 | 2rexbidv 2460 | . . . . . . . . . 10 |
27 | txval.1 | . . . . . . . . . . 11 | |
28 | eqid 2139 | . . . . . . . . . . . 12 | |
29 | 28 | rnmpo 5881 | . . . . . . . . . . 11 |
30 | 27, 29 | eqtri 2160 | . . . . . . . . . 10 |
31 | 24, 26, 30 | elab2 2832 | . . . . . . . . 9 |
32 | 21, 31 | sylibr 133 | . . . . . . . 8 |
33 | elssuni 3764 | . . . . . . . 8 | |
34 | 32, 33 | syl 14 | . . . . . . 7 |
35 | 34 | sseld 3096 | . . . . . 6 |
36 | 14, 35 | syl5bir 152 | . . . . 5 |
37 | 36 | rexlimivv 2555 | . . . 4 |
38 | 13, 37 | sylbi 120 | . . 3 |
39 | 1, 38 | relssi 4630 | . 2 |
40 | elssuni 3764 | . . . . . . . . . 10 | |
41 | 40, 2 | sseqtrrdi 3146 | . . . . . . . . 9 |
42 | elssuni 3764 | . . . . . . . . . 10 | |
43 | 42, 6 | sseqtrrdi 3146 | . . . . . . . . 9 |
44 | xpss12 4646 | . . . . . . . . 9 | |
45 | 41, 43, 44 | syl2an 287 | . . . . . . . 8 |
46 | vex 2689 | . . . . . . . . . 10 | |
47 | vex 2689 | . . . . . . . . . 10 | |
48 | 46, 47 | xpex 4654 | . . . . . . . . 9 |
49 | 48 | elpw 3516 | . . . . . . . 8 |
50 | 45, 49 | sylibr 133 | . . . . . . 7 |
51 | 50 | rgen2 2518 | . . . . . 6 |
52 | 28 | fmpo 6099 | . . . . . 6 |
53 | 51, 52 | mpbi 144 | . . . . 5 |
54 | frn 5281 | . . . . 5 | |
55 | 53, 54 | ax-mp 5 | . . . 4 |
56 | 27, 55 | eqsstri 3129 | . . 3 |
57 | sspwuni 3897 | . . 3 | |
58 | 56, 57 | mpbi 144 | . 2 |
59 | 39, 58 | eqssi 3113 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1331 wcel 1480 cab 2125 wral 2416 wrex 2417 wss 3071 cpw 3510 cop 3530 cuni 3736 cxp 4537 crn 4540 wf 5119 cmpo 5776 |
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-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 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 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-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 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-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: txbasex 12426 txtopon 12431 |
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