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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 2157 | . . . . 5 | |
2 | eqid 2157 | . . . . 5 | |
3 | 1, 2 | txcnmpt 12673 | . . . 4 |
4 | uptx.1 | . . . . 5 | |
5 | 4 | oveq2i 5835 | . . . 4 |
6 | 3, 5 | eleqtrrdi 2251 | . . 3 |
7 | uptx.2 | . . . . . 6 | |
8 | 1, 7 | cnf 12604 | . . . . 5 |
9 | uptx.3 | . . . . . 6 | |
10 | 1, 9 | cnf 12604 | . . . . 5 |
11 | ffn 5319 | . . . . . . . 8 | |
12 | 11 | adantr 274 | . . . . . . 7 |
13 | fo1st 6105 | . . . . . . . . . 10 | |
14 | fofn 5394 | . . . . . . . . . 10 | |
15 | 13, 14 | ax-mp 5 | . . . . . . . . 9 |
16 | ssv 3150 | . . . . . . . . 9 | |
17 | fnssres 5283 | . . . . . . . . 9 | |
18 | 15, 16, 17 | mp2an 423 | . . . . . . . 8 |
19 | ffvelrn 5600 | . . . . . . . . . . . 12 | |
20 | ffvelrn 5600 | . . . . . . . . . . . 12 | |
21 | opelxpi 4618 | . . . . . . . . . . . 12 | |
22 | 19, 20, 21 | syl2an 287 | . . . . . . . . . . 11 |
23 | 22 | anandirs 583 | . . . . . . . . . 10 |
24 | 23 | fmpttd 5622 | . . . . . . . . 9 |
25 | 24 | ffnd 5320 | . . . . . . . 8 |
26 | 24 | frnd 5329 | . . . . . . . 8 |
27 | fnco 5278 | . . . . . . . 8 | |
28 | 18, 25, 26, 27 | mp3an2i 1324 | . . . . . . 7 |
29 | fvco3 5539 | . . . . . . . . 9 | |
30 | 24, 29 | sylan 281 | . . . . . . . 8 |
31 | fveq2 5468 | . . . . . . . . . . 11 | |
32 | fveq2 5468 | . . . . . . . . . . 11 | |
33 | 31, 32 | opeq12d 3749 | . . . . . . . . . 10 |
34 | simpr 109 | . . . . . . . . . 10 | |
35 | simpll 519 | . . . . . . . . . . . 12 | |
36 | 35, 34 | ffvelrnd 5603 | . . . . . . . . . . 11 |
37 | simplr 520 | . . . . . . . . . . . 12 | |
38 | 37, 34 | ffvelrnd 5603 | . . . . . . . . . . 11 |
39 | 36, 38 | opelxpd 4619 | . . . . . . . . . 10 |
40 | 2, 33, 34, 39 | fvmptd3 5561 | . . . . . . . . 9 |
41 | 40 | fveq2d 5472 | . . . . . . . 8 |
42 | ffvelrn 5600 | . . . . . . . . . . . 12 | |
43 | ffvelrn 5600 | . . . . . . . . . . . 12 | |
44 | opelxpi 4618 | . . . . . . . . . . . 12 | |
45 | 42, 43, 44 | syl2an 287 | . . . . . . . . . . 11 |
46 | 45 | anandirs 583 | . . . . . . . . . 10 |
47 | 46 | fvresd 5493 | . . . . . . . . 9 |
48 | op1stg 6098 | . . . . . . . . . 10 | |
49 | 36, 38, 48 | syl2anc 409 | . . . . . . . . 9 |
50 | 47, 49 | eqtrd 2190 | . . . . . . . 8 |
51 | 30, 41, 50 | 3eqtrrd 2195 | . . . . . . 7 |
52 | 12, 28, 51 | eqfnfvd 5568 | . . . . . 6 |
53 | uptx.5 | . . . . . . . 8 | |
54 | uptx.4 | . . . . . . . . 9 | |
55 | 54 | reseq2i 4863 | . . . . . . . 8 |
56 | 53, 55 | eqtri 2178 | . . . . . . 7 |
57 | 56 | coeq1i 4745 | . . . . . 6 |
58 | 52, 57 | eqtr4di 2208 | . . . . 5 |
59 | 8, 10, 58 | syl2an 287 | . . . 4 |
60 | ffn 5319 | . . . . . . . 8 | |
61 | 60 | adantl 275 | . . . . . . 7 |
62 | fo2nd 6106 | . . . . . . . . . 10 | |
63 | fofn 5394 | . . . . . . . . . 10 | |
64 | 62, 63 | ax-mp 5 | . . . . . . . . 9 |
65 | fnssres 5283 | . . . . . . . . 9 | |
66 | 64, 16, 65 | mp2an 423 | . . . . . . . 8 |
67 | fnco 5278 | . . . . . . . 8 | |
68 | 66, 25, 26, 67 | mp3an2i 1324 | . . . . . . 7 |
69 | fvco3 5539 | . . . . . . . . 9 | |
70 | 24, 69 | sylan 281 | . . . . . . . 8 |
71 | 40 | fveq2d 5472 | . . . . . . . 8 |
72 | 46 | fvresd 5493 | . . . . . . . . 9 |
73 | op2ndg 6099 | . . . . . . . . . 10 | |
74 | 36, 38, 73 | syl2anc 409 | . . . . . . . . 9 |
75 | 72, 74 | eqtrd 2190 | . . . . . . . 8 |
76 | 70, 71, 75 | 3eqtrrd 2195 | . . . . . . 7 |
77 | 61, 68, 76 | eqfnfvd 5568 | . . . . . 6 |
78 | uptx.6 | . . . . . . . 8 | |
79 | 54 | reseq2i 4863 | . . . . . . . 8 |
80 | 78, 79 | eqtri 2178 | . . . . . . 7 |
81 | 80 | coeq1i 4745 | . . . . . 6 |
82 | 77, 81 | eqtr4di 2208 | . . . . 5 |
83 | 8, 10, 82 | syl2an 287 | . . . 4 |
84 | 6, 59, 83 | jca32 308 | . . 3 |
85 | eleq1 2220 | . . . . 5 | |
86 | coeq2 4744 | . . . . . . 7 | |
87 | 86 | eqeq2d 2169 | . . . . . 6 |
88 | coeq2 4744 | . . . . . . 7 | |
89 | 88 | eqeq2d 2169 | . . . . . 6 |
90 | 87, 89 | anbi12d 465 | . . . . 5 |
91 | 85, 90 | anbi12d 465 | . . . 4 |
92 | 91 | spcegv 2800 | . . 3 |
93 | 6, 84, 92 | sylc 62 | . 2 |
94 | eqid 2157 | . . . . . . . 8 | |
95 | 1, 94 | cnf 12604 | . . . . . . 7 |
96 | cntop2 12602 | . . . . . . . . 9 | |
97 | cntop2 12602 | . . . . . . . . 9 | |
98 | 4 | unieqi 3782 | . . . . . . . . . 10 |
99 | 7, 9 | txuni 12663 | . . . . . . . . . 10 |
100 | 98, 99 | eqtr4id 2209 | . . . . . . . . 9 |
101 | 96, 97, 100 | syl2an 287 | . . . . . . . 8 |
102 | 101 | feq3d 5308 | . . . . . . 7 |
103 | 95, 102 | syl5ib 153 | . . . . . 6 |
104 | 103 | anim1d 334 | . . . . 5 |
105 | 3anass 967 | . . . . 5 | |
106 | 104, 105 | syl6ibr 161 | . . . 4 |
107 | 106 | alrimiv 1854 | . . 3 |
108 | cntop1 12601 | . . . . . 6 | |
109 | uniexg 4399 | . . . . . 6 | |
110 | 108, 109 | syl 14 | . . . . 5 |
111 | 56, 80 | upxp 12672 | . . . . 5 |
112 | 110, 8, 10, 111 | syl2an3an 1280 | . . . 4 |
113 | eumo 2038 | . . . 4 | |
114 | 112, 113 | syl 14 | . . 3 |
115 | moim 2070 | . . 3 | |
116 | 107, 114, 115 | sylc 62 | . 2 |
117 | df-reu 2442 | . . 3 | |
118 | eu5 2053 | . . 3 | |
119 | 117, 118 | bitri 183 | . 2 |
120 | 93, 116, 119 | sylanbrc 414 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 963 wal 1333 wceq 1335 wex 1472 weu 2006 wmo 2007 wcel 2128 wreu 2437 cvv 2712 wss 3102 cop 3563 cuni 3772 cmpt 4025 cxp 4584 crn 4587 cres 4588 ccom 4590 wfn 5165 wf 5166 wfo 5168 cfv 5170 (class class class)co 5824 c1st 6086 c2nd 6087 ctop 12395 ccn 12585 ctx 12652 |
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 604 ax-in2 605 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-coll 4079 ax-sep 4082 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 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-ne 2328 df-ral 2440 df-rex 2441 df-reu 2442 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-id 4253 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-res 4598 df-ima 4599 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-ov 5827 df-oprab 5828 df-mpo 5829 df-1st 6088 df-2nd 6089 df-map 6595 df-topgen 12372 df-top 12396 df-topon 12409 df-bases 12441 df-cn 12588 df-tx 12653 |
This theorem is referenced by: txcn 12675 |
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