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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 4720 | . . 3 | |
2 | txuni2.1 | . . . . . . . 8 | |
3 | 2 | eleq2i 2237 | . . . . . . 7 |
4 | eluni2 3800 | . . . . . . 7 | |
5 | 3, 4 | bitri 183 | . . . . . 6 |
6 | txuni2.2 | . . . . . . . 8 | |
7 | 6 | eleq2i 2237 | . . . . . . 7 |
8 | eluni2 3800 | . . . . . . 7 | |
9 | 7, 8 | bitri 183 | . . . . . 6 |
10 | 5, 9 | anbi12i 457 | . . . . 5 |
11 | opelxp 4641 | . . . . 5 | |
12 | reeanv 2639 | . . . . 5 | |
13 | 10, 11, 12 | 3bitr4i 211 | . . . 4 |
14 | opelxp 4641 | . . . . . 6 | |
15 | eqid 2170 | . . . . . . . . . 10 | |
16 | xpeq1 4625 | . . . . . . . . . . . 12 | |
17 | 16 | eqeq2d 2182 | . . . . . . . . . . 11 |
18 | xpeq2 4626 | . . . . . . . . . . . 12 | |
19 | 18 | eqeq2d 2182 | . . . . . . . . . . 11 |
20 | 17, 19 | rspc2ev 2849 | . . . . . . . . . 10 |
21 | 15, 20 | mp3an3 1321 | . . . . . . . . 9 |
22 | vex 2733 | . . . . . . . . . . 11 | |
23 | vex 2733 | . . . . . . . . . . 11 | |
24 | 22, 23 | xpex 4726 | . . . . . . . . . 10 |
25 | eqeq1 2177 | . . . . . . . . . . 11 | |
26 | 25 | 2rexbidv 2495 | . . . . . . . . . 10 |
27 | txval.1 | . . . . . . . . . . 11 | |
28 | eqid 2170 | . . . . . . . . . . . 12 | |
29 | 28 | rnmpo 5963 | . . . . . . . . . . 11 |
30 | 27, 29 | eqtri 2191 | . . . . . . . . . 10 |
31 | 24, 26, 30 | elab2 2878 | . . . . . . . . 9 |
32 | 21, 31 | sylibr 133 | . . . . . . . 8 |
33 | elssuni 3824 | . . . . . . . 8 | |
34 | 32, 33 | syl 14 | . . . . . . 7 |
35 | 34 | sseld 3146 | . . . . . 6 |
36 | 14, 35 | syl5bir 152 | . . . . 5 |
37 | 36 | rexlimivv 2593 | . . . 4 |
38 | 13, 37 | sylbi 120 | . . 3 |
39 | 1, 38 | relssi 4702 | . 2 |
40 | elssuni 3824 | . . . . . . . . . 10 | |
41 | 40, 2 | sseqtrrdi 3196 | . . . . . . . . 9 |
42 | elssuni 3824 | . . . . . . . . . 10 | |
43 | 42, 6 | sseqtrrdi 3196 | . . . . . . . . 9 |
44 | xpss12 4718 | . . . . . . . . 9 | |
45 | 41, 43, 44 | syl2an 287 | . . . . . . . 8 |
46 | vex 2733 | . . . . . . . . . 10 | |
47 | vex 2733 | . . . . . . . . . 10 | |
48 | 46, 47 | xpex 4726 | . . . . . . . . 9 |
49 | 48 | elpw 3572 | . . . . . . . 8 |
50 | 45, 49 | sylibr 133 | . . . . . . 7 |
51 | 50 | rgen2 2556 | . . . . . 6 |
52 | 28 | fmpo 6180 | . . . . . 6 |
53 | 51, 52 | mpbi 144 | . . . . 5 |
54 | frn 5356 | . . . . 5 | |
55 | 53, 54 | ax-mp 5 | . . . 4 |
56 | 27, 55 | eqsstri 3179 | . . 3 |
57 | sspwuni 3957 | . . 3 | |
58 | 56, 57 | mpbi 144 | . 2 |
59 | 39, 58 | eqssi 3163 | 1 |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1348 wcel 2141 cab 2156 wral 2448 wrex 2449 wss 3121 cpw 3566 cop 3586 cuni 3796 cxp 4609 crn 4612 wf 5194 cmpo 5855 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-iun 3875 df-br 3990 df-opab 4051 df-mpt 4052 df-id 4278 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 |
This theorem is referenced by: txbasex 13051 txtopon 13056 |
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