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Mirrors > Home > ILE Home > Th. List > uptx | Unicode version |
Description: Universal property of the binary topological product. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.) |
Ref | Expression |
---|---|
uptx.1 | |
uptx.2 | |
uptx.3 | |
uptx.4 | |
uptx.5 | |
uptx.6 |
Ref | Expression |
---|---|
uptx |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2139 | . . . . 5 | |
2 | eqid 2139 | . . . . 5 | |
3 | 1, 2 | txcnmpt 12442 | . . . 4 |
4 | uptx.1 | . . . . 5 | |
5 | 4 | oveq2i 5785 | . . . 4 |
6 | 3, 5 | eleqtrrdi 2233 | . . 3 |
7 | uptx.2 | . . . . . 6 | |
8 | 1, 7 | cnf 12373 | . . . . 5 |
9 | uptx.3 | . . . . . 6 | |
10 | 1, 9 | cnf 12373 | . . . . 5 |
11 | ffn 5272 | . . . . . . . 8 | |
12 | 11 | adantr 274 | . . . . . . 7 |
13 | fo1st 6055 | . . . . . . . . . 10 | |
14 | fofn 5347 | . . . . . . . . . 10 | |
15 | 13, 14 | ax-mp 5 | . . . . . . . . 9 |
16 | ssv 3119 | . . . . . . . . 9 | |
17 | fnssres 5236 | . . . . . . . . 9 | |
18 | 15, 16, 17 | mp2an 422 | . . . . . . . 8 |
19 | ffvelrn 5553 | . . . . . . . . . . . 12 | |
20 | ffvelrn 5553 | . . . . . . . . . . . 12 | |
21 | opelxpi 4571 | . . . . . . . . . . . 12 | |
22 | 19, 20, 21 | syl2an 287 | . . . . . . . . . . 11 |
23 | 22 | anandirs 582 | . . . . . . . . . 10 |
24 | 23 | fmpttd 5575 | . . . . . . . . 9 |
25 | 24 | ffnd 5273 | . . . . . . . 8 |
26 | 24 | frnd 5282 | . . . . . . . 8 |
27 | fnco 5231 | . . . . . . . 8 | |
28 | 18, 25, 26, 27 | mp3an2i 1320 | . . . . . . 7 |
29 | fvco3 5492 | . . . . . . . . 9 | |
30 | 24, 29 | sylan 281 | . . . . . . . 8 |
31 | fveq2 5421 | . . . . . . . . . . 11 | |
32 | fveq2 5421 | . . . . . . . . . . 11 | |
33 | 31, 32 | opeq12d 3713 | . . . . . . . . . 10 |
34 | simpr 109 | . . . . . . . . . 10 | |
35 | simpll 518 | . . . . . . . . . . . 12 | |
36 | 35, 34 | ffvelrnd 5556 | . . . . . . . . . . 11 |
37 | simplr 519 | . . . . . . . . . . . 12 | |
38 | 37, 34 | ffvelrnd 5556 | . . . . . . . . . . 11 |
39 | 36, 38 | opelxpd 4572 | . . . . . . . . . 10 |
40 | 2, 33, 34, 39 | fvmptd3 5514 | . . . . . . . . 9 |
41 | 40 | fveq2d 5425 | . . . . . . . 8 |
42 | ffvelrn 5553 | . . . . . . . . . . . 12 | |
43 | ffvelrn 5553 | . . . . . . . . . . . 12 | |
44 | opelxpi 4571 | . . . . . . . . . . . 12 | |
45 | 42, 43, 44 | syl2an 287 | . . . . . . . . . . 11 |
46 | 45 | anandirs 582 | . . . . . . . . . 10 |
47 | 46 | fvresd 5446 | . . . . . . . . 9 |
48 | op1stg 6048 | . . . . . . . . . 10 | |
49 | 36, 38, 48 | syl2anc 408 | . . . . . . . . 9 |
50 | 47, 49 | eqtrd 2172 | . . . . . . . 8 |
51 | 30, 41, 50 | 3eqtrrd 2177 | . . . . . . 7 |
52 | 12, 28, 51 | eqfnfvd 5521 | . . . . . 6 |
53 | uptx.5 | . . . . . . . 8 | |
54 | uptx.4 | . . . . . . . . 9 | |
55 | 54 | reseq2i 4816 | . . . . . . . 8 |
56 | 53, 55 | eqtri 2160 | . . . . . . 7 |
57 | 56 | coeq1i 4698 | . . . . . 6 |
58 | 52, 57 | syl6eqr 2190 | . . . . 5 |
59 | 8, 10, 58 | syl2an 287 | . . . 4 |
60 | ffn 5272 | . . . . . . . 8 | |
61 | 60 | adantl 275 | . . . . . . 7 |
62 | fo2nd 6056 | . . . . . . . . . 10 | |
63 | fofn 5347 | . . . . . . . . . 10 | |
64 | 62, 63 | ax-mp 5 | . . . . . . . . 9 |
65 | fnssres 5236 | . . . . . . . . 9 | |
66 | 64, 16, 65 | mp2an 422 | . . . . . . . 8 |
67 | fnco 5231 | . . . . . . . 8 | |
68 | 66, 25, 26, 67 | mp3an2i 1320 | . . . . . . 7 |
69 | fvco3 5492 | . . . . . . . . 9 | |
70 | 24, 69 | sylan 281 | . . . . . . . 8 |
71 | 40 | fveq2d 5425 | . . . . . . . 8 |
72 | 46 | fvresd 5446 | . . . . . . . . 9 |
73 | op2ndg 6049 | . . . . . . . . . 10 | |
74 | 36, 38, 73 | syl2anc 408 | . . . . . . . . 9 |
75 | 72, 74 | eqtrd 2172 | . . . . . . . 8 |
76 | 70, 71, 75 | 3eqtrrd 2177 | . . . . . . 7 |
77 | 61, 68, 76 | eqfnfvd 5521 | . . . . . 6 |
78 | uptx.6 | . . . . . . . 8 | |
79 | 54 | reseq2i 4816 | . . . . . . . 8 |
80 | 78, 79 | eqtri 2160 | . . . . . . 7 |
81 | 80 | coeq1i 4698 | . . . . . 6 |
82 | 77, 81 | syl6eqr 2190 | . . . . 5 |
83 | 8, 10, 82 | syl2an 287 | . . . 4 |
84 | 6, 59, 83 | jca32 308 | . . 3 |
85 | eleq1 2202 | . . . . 5 | |
86 | coeq2 4697 | . . . . . . 7 | |
87 | 86 | eqeq2d 2151 | . . . . . 6 |
88 | coeq2 4697 | . . . . . . 7 | |
89 | 88 | eqeq2d 2151 | . . . . . 6 |
90 | 87, 89 | anbi12d 464 | . . . . 5 |
91 | 85, 90 | anbi12d 464 | . . . 4 |
92 | 91 | spcegv 2774 | . . 3 |
93 | 6, 84, 92 | sylc 62 | . 2 |
94 | eqid 2139 | . . . . . . . 8 | |
95 | 1, 94 | cnf 12373 | . . . . . . 7 |
96 | cntop2 12371 | . . . . . . . . 9 | |
97 | cntop2 12371 | . . . . . . . . 9 | |
98 | 7, 9 | txuni 12432 | . . . . . . . . . 10 |
99 | 4 | unieqi 3746 | . . . . . . . . . 10 |
100 | 98, 99 | syl6reqr 2191 | . . . . . . . . 9 |
101 | 96, 97, 100 | syl2an 287 | . . . . . . . 8 |
102 | 101 | feq3d 5261 | . . . . . . 7 |
103 | 95, 102 | syl5ib 153 | . . . . . 6 |
104 | 103 | anim1d 334 | . . . . 5 |
105 | 3anass 966 | . . . . 5 | |
106 | 104, 105 | syl6ibr 161 | . . . 4 |
107 | 106 | alrimiv 1846 | . . 3 |
108 | cntop1 12370 | . . . . . 6 | |
109 | uniexg 4361 | . . . . . 6 | |
110 | 108, 109 | syl 14 | . . . . 5 |
111 | 56, 80 | upxp 12441 | . . . . 5 |
112 | 110, 8, 10, 111 | syl2an3an 1276 | . . . 4 |
113 | eumo 2031 | . . . 4 | |
114 | 112, 113 | syl 14 | . . 3 |
115 | moim 2063 | . . 3 | |
116 | 107, 114, 115 | sylc 62 | . 2 |
117 | df-reu 2423 | . . 3 | |
118 | eu5 2046 | . . 3 | |
119 | 117, 118 | bitri 183 | . 2 |
120 | 93, 116, 119 | sylanbrc 413 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 962 wal 1329 wceq 1331 wex 1468 wcel 1480 weu 1999 wmo 2000 wreu 2418 cvv 2686 wss 3071 cop 3530 cuni 3736 cmpt 3989 cxp 4537 crn 4540 cres 4541 ccom 4543 wfn 5118 wf 5119 wfo 5121 cfv 5123 (class class class)co 5774 c1st 6036 c2nd 6037 ctop 12164 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-nul 3364 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 |
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