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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 4698 | . . 3 | |
2 | txuni2.1 | . . . . . . . 8 | |
3 | 2 | eleq2i 2224 | . . . . . . 7 |
4 | eluni2 3778 | . . . . . . 7 | |
5 | 3, 4 | bitri 183 | . . . . . 6 |
6 | txuni2.2 | . . . . . . . 8 | |
7 | 6 | eleq2i 2224 | . . . . . . 7 |
8 | eluni2 3778 | . . . . . . 7 | |
9 | 7, 8 | bitri 183 | . . . . . 6 |
10 | 5, 9 | anbi12i 456 | . . . . 5 |
11 | opelxp 4619 | . . . . 5 | |
12 | reeanv 2626 | . . . . 5 | |
13 | 10, 11, 12 | 3bitr4i 211 | . . . 4 |
14 | opelxp 4619 | . . . . . 6 | |
15 | eqid 2157 | . . . . . . . . . 10 | |
16 | xpeq1 4603 | . . . . . . . . . . . 12 | |
17 | 16 | eqeq2d 2169 | . . . . . . . . . . 11 |
18 | xpeq2 4604 | . . . . . . . . . . . 12 | |
19 | 18 | eqeq2d 2169 | . . . . . . . . . . 11 |
20 | 17, 19 | rspc2ev 2831 | . . . . . . . . . 10 |
21 | 15, 20 | mp3an3 1308 | . . . . . . . . 9 |
22 | vex 2715 | . . . . . . . . . . 11 | |
23 | vex 2715 | . . . . . . . . . . 11 | |
24 | 22, 23 | xpex 4704 | . . . . . . . . . 10 |
25 | eqeq1 2164 | . . . . . . . . . . 11 | |
26 | 25 | 2rexbidv 2482 | . . . . . . . . . 10 |
27 | txval.1 | . . . . . . . . . . 11 | |
28 | eqid 2157 | . . . . . . . . . . . 12 | |
29 | 28 | rnmpo 5934 | . . . . . . . . . . 11 |
30 | 27, 29 | eqtri 2178 | . . . . . . . . . 10 |
31 | 24, 26, 30 | elab2 2860 | . . . . . . . . 9 |
32 | 21, 31 | sylibr 133 | . . . . . . . 8 |
33 | elssuni 3802 | . . . . . . . 8 | |
34 | 32, 33 | syl 14 | . . . . . . 7 |
35 | 34 | sseld 3127 | . . . . . 6 |
36 | 14, 35 | syl5bir 152 | . . . . 5 |
37 | 36 | rexlimivv 2580 | . . . 4 |
38 | 13, 37 | sylbi 120 | . . 3 |
39 | 1, 38 | relssi 4680 | . 2 |
40 | elssuni 3802 | . . . . . . . . . 10 | |
41 | 40, 2 | sseqtrrdi 3177 | . . . . . . . . 9 |
42 | elssuni 3802 | . . . . . . . . . 10 | |
43 | 42, 6 | sseqtrrdi 3177 | . . . . . . . . 9 |
44 | xpss12 4696 | . . . . . . . . 9 | |
45 | 41, 43, 44 | syl2an 287 | . . . . . . . 8 |
46 | vex 2715 | . . . . . . . . . 10 | |
47 | vex 2715 | . . . . . . . . . 10 | |
48 | 46, 47 | xpex 4704 | . . . . . . . . 9 |
49 | 48 | elpw 3550 | . . . . . . . 8 |
50 | 45, 49 | sylibr 133 | . . . . . . 7 |
51 | 50 | rgen2 2543 | . . . . . 6 |
52 | 28 | fmpo 6152 | . . . . . 6 |
53 | 51, 52 | mpbi 144 | . . . . 5 |
54 | frn 5331 | . . . . 5 | |
55 | 53, 54 | ax-mp 5 | . . . 4 |
56 | 27, 55 | eqsstri 3160 | . . 3 |
57 | sspwuni 3935 | . . 3 | |
58 | 56, 57 | mpbi 144 | . 2 |
59 | 39, 58 | eqssi 3144 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1335 wcel 2128 cab 2143 wral 2435 wrex 2436 wss 3102 cpw 3544 cop 3564 cuni 3774 cxp 4587 crn 4590 wf 5169 cmpo 5829 |
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 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-pow 4138 ax-pr 4172 ax-un 4396 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-un 3106 df-in 3108 df-ss 3115 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-iun 3853 df-br 3968 df-opab 4029 df-mpt 4030 df-id 4256 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-fv 5181 df-oprab 5831 df-mpo 5832 df-1st 6091 df-2nd 6092 |
This theorem is referenced by: txbasex 12753 txtopon 12758 |
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