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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 2170 | . . . . 5 | |
2 | eqid 2170 | . . . . 5 | |
3 | 1, 2 | txcnmpt 13067 | . . . 4 |
4 | uptx.1 | . . . . 5 | |
5 | 4 | oveq2i 5864 | . . . 4 |
6 | 3, 5 | eleqtrrdi 2264 | . . 3 |
7 | uptx.2 | . . . . . 6 | |
8 | 1, 7 | cnf 12998 | . . . . 5 |
9 | uptx.3 | . . . . . 6 | |
10 | 1, 9 | cnf 12998 | . . . . 5 |
11 | ffn 5347 | . . . . . . . 8 | |
12 | 11 | adantr 274 | . . . . . . 7 |
13 | fo1st 6136 | . . . . . . . . . 10 | |
14 | fofn 5422 | . . . . . . . . . 10 | |
15 | 13, 14 | ax-mp 5 | . . . . . . . . 9 |
16 | ssv 3169 | . . . . . . . . 9 | |
17 | fnssres 5311 | . . . . . . . . 9 | |
18 | 15, 16, 17 | mp2an 424 | . . . . . . . 8 |
19 | ffvelrn 5629 | . . . . . . . . . . . 12 | |
20 | ffvelrn 5629 | . . . . . . . . . . . 12 | |
21 | opelxpi 4643 | . . . . . . . . . . . 12 | |
22 | 19, 20, 21 | syl2an 287 | . . . . . . . . . . 11 |
23 | 22 | anandirs 588 | . . . . . . . . . 10 |
24 | 23 | fmpttd 5651 | . . . . . . . . 9 |
25 | 24 | ffnd 5348 | . . . . . . . 8 |
26 | 24 | frnd 5357 | . . . . . . . 8 |
27 | fnco 5306 | . . . . . . . 8 | |
28 | 18, 25, 26, 27 | mp3an2i 1337 | . . . . . . 7 |
29 | fvco3 5567 | . . . . . . . . 9 | |
30 | 24, 29 | sylan 281 | . . . . . . . 8 |
31 | fveq2 5496 | . . . . . . . . . . 11 | |
32 | fveq2 5496 | . . . . . . . . . . 11 | |
33 | 31, 32 | opeq12d 3773 | . . . . . . . . . 10 |
34 | simpr 109 | . . . . . . . . . 10 | |
35 | simpll 524 | . . . . . . . . . . . 12 | |
36 | 35, 34 | ffvelrnd 5632 | . . . . . . . . . . 11 |
37 | simplr 525 | . . . . . . . . . . . 12 | |
38 | 37, 34 | ffvelrnd 5632 | . . . . . . . . . . 11 |
39 | 36, 38 | opelxpd 4644 | . . . . . . . . . 10 |
40 | 2, 33, 34, 39 | fvmptd3 5589 | . . . . . . . . 9 |
41 | 40 | fveq2d 5500 | . . . . . . . 8 |
42 | ffvelrn 5629 | . . . . . . . . . . . 12 | |
43 | ffvelrn 5629 | . . . . . . . . . . . 12 | |
44 | opelxpi 4643 | . . . . . . . . . . . 12 | |
45 | 42, 43, 44 | syl2an 287 | . . . . . . . . . . 11 |
46 | 45 | anandirs 588 | . . . . . . . . . 10 |
47 | 46 | fvresd 5521 | . . . . . . . . 9 |
48 | op1stg 6129 | . . . . . . . . . 10 | |
49 | 36, 38, 48 | syl2anc 409 | . . . . . . . . 9 |
50 | 47, 49 | eqtrd 2203 | . . . . . . . 8 |
51 | 30, 41, 50 | 3eqtrrd 2208 | . . . . . . 7 |
52 | 12, 28, 51 | eqfnfvd 5596 | . . . . . 6 |
53 | uptx.5 | . . . . . . . 8 | |
54 | uptx.4 | . . . . . . . . 9 | |
55 | 54 | reseq2i 4888 | . . . . . . . 8 |
56 | 53, 55 | eqtri 2191 | . . . . . . 7 |
57 | 56 | coeq1i 4770 | . . . . . 6 |
58 | 52, 57 | eqtr4di 2221 | . . . . 5 |
59 | 8, 10, 58 | syl2an 287 | . . . 4 |
60 | ffn 5347 | . . . . . . . 8 | |
61 | 60 | adantl 275 | . . . . . . 7 |
62 | fo2nd 6137 | . . . . . . . . . 10 | |
63 | fofn 5422 | . . . . . . . . . 10 | |
64 | 62, 63 | ax-mp 5 | . . . . . . . . 9 |
65 | fnssres 5311 | . . . . . . . . 9 | |
66 | 64, 16, 65 | mp2an 424 | . . . . . . . 8 |
67 | fnco 5306 | . . . . . . . 8 | |
68 | 66, 25, 26, 67 | mp3an2i 1337 | . . . . . . 7 |
69 | fvco3 5567 | . . . . . . . . 9 | |
70 | 24, 69 | sylan 281 | . . . . . . . 8 |
71 | 40 | fveq2d 5500 | . . . . . . . 8 |
72 | 46 | fvresd 5521 | . . . . . . . . 9 |
73 | op2ndg 6130 | . . . . . . . . . 10 | |
74 | 36, 38, 73 | syl2anc 409 | . . . . . . . . 9 |
75 | 72, 74 | eqtrd 2203 | . . . . . . . 8 |
76 | 70, 71, 75 | 3eqtrrd 2208 | . . . . . . 7 |
77 | 61, 68, 76 | eqfnfvd 5596 | . . . . . 6 |
78 | uptx.6 | . . . . . . . 8 | |
79 | 54 | reseq2i 4888 | . . . . . . . 8 |
80 | 78, 79 | eqtri 2191 | . . . . . . 7 |
81 | 80 | coeq1i 4770 | . . . . . 6 |
82 | 77, 81 | eqtr4di 2221 | . . . . 5 |
83 | 8, 10, 82 | syl2an 287 | . . . 4 |
84 | 6, 59, 83 | jca32 308 | . . 3 |
85 | eleq1 2233 | . . . . 5 | |
86 | coeq2 4769 | . . . . . . 7 | |
87 | 86 | eqeq2d 2182 | . . . . . 6 |
88 | coeq2 4769 | . . . . . . 7 | |
89 | 88 | eqeq2d 2182 | . . . . . 6 |
90 | 87, 89 | anbi12d 470 | . . . . 5 |
91 | 85, 90 | anbi12d 470 | . . . 4 |
92 | 91 | spcegv 2818 | . . 3 |
93 | 6, 84, 92 | sylc 62 | . 2 |
94 | eqid 2170 | . . . . . . . 8 | |
95 | 1, 94 | cnf 12998 | . . . . . . 7 |
96 | cntop2 12996 | . . . . . . . . 9 | |
97 | cntop2 12996 | . . . . . . . . 9 | |
98 | 4 | unieqi 3806 | . . . . . . . . . 10 |
99 | 7, 9 | txuni 13057 | . . . . . . . . . 10 |
100 | 98, 99 | eqtr4id 2222 | . . . . . . . . 9 |
101 | 96, 97, 100 | syl2an 287 | . . . . . . . 8 |
102 | 101 | feq3d 5336 | . . . . . . 7 |
103 | 95, 102 | syl5ib 153 | . . . . . 6 |
104 | 103 | anim1d 334 | . . . . 5 |
105 | 3anass 977 | . . . . 5 | |
106 | 104, 105 | syl6ibr 161 | . . . 4 |
107 | 106 | alrimiv 1867 | . . 3 |
108 | cntop1 12995 | . . . . . 6 | |
109 | uniexg 4424 | . . . . . 6 | |
110 | 108, 109 | syl 14 | . . . . 5 |
111 | 56, 80 | upxp 13066 | . . . . 5 |
112 | 110, 8, 10, 111 | syl2an3an 1293 | . . . 4 |
113 | eumo 2051 | . . . 4 | |
114 | 112, 113 | syl 14 | . . 3 |
115 | moim 2083 | . . 3 | |
116 | 107, 114, 115 | sylc 62 | . 2 |
117 | df-reu 2455 | . . 3 | |
118 | eu5 2066 | . . 3 | |
119 | 117, 118 | bitri 183 | . 2 |
120 | 93, 116, 119 | sylanbrc 415 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 w3a 973 wal 1346 wceq 1348 wex 1485 weu 2019 wmo 2020 wcel 2141 wreu 2450 cvv 2730 wss 3121 cop 3586 cuni 3796 cmpt 4050 cxp 4609 crn 4612 cres 4613 ccom 4615 wfn 5193 wf 5194 wfo 5196 cfv 5198 (class class class)co 5853 c1st 6117 c2nd 6118 ctop 12789 ccn 12979 ctx 13046 |
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 609 ax-in2 610 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-coll 4104 ax-sep 4107 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 |
This theorem depends on definitions: df-bi 116 df-3an 975 df-tru 1351 df-fal 1354 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-ne 2341 df-ral 2453 df-rex 2454 df-reu 2455 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 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-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1st 6119 df-2nd 6120 df-map 6628 df-topgen 12600 df-top 12790 df-topon 12803 df-bases 12835 df-cn 12982 df-tx 13047 |
This theorem is referenced by: txcn 13069 |
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