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Mirrors > Home > NFE Home > Th. List > brtxp | Unicode version |
Description: Binary relationship over a tail cross product. (Contributed by SF, 11-Feb-2015.) |
Ref | Expression |
---|---|
brtxp |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | brin 4693 | . . . 4 | |
2 | brco 4883 | . . . . 5 | |
3 | brco 4883 | . . . . 5 | |
4 | 2, 3 | anbi12i 678 | . . . 4 |
5 | 1, 4 | bitri 240 | . . 3 |
6 | df-txp 5736 | . . . 4 | |
7 | 6 | breqi 4645 | . . 3 |
8 | eeanv 1913 | . . 3 | |
9 | 5, 7, 8 | 3bitr4i 268 | . 2 |
10 | an4 797 | . . . 4 | |
11 | ancom 437 | . . . . 5 | |
12 | brcnv 4892 | . . . . . . . . 9 | |
13 | vex 2862 | . . . . . . . . . 10 | |
14 | 13 | br1st 4858 | . . . . . . . . 9 |
15 | 12, 14 | bitri 240 | . . . . . . . 8 |
16 | brcnv 4892 | . . . . . . . . 9 | |
17 | vex 2862 | . . . . . . . . . 10 | |
18 | 17 | br2nd 4859 | . . . . . . . . 9 |
19 | 16, 18 | bitri 240 | . . . . . . . 8 |
20 | 15, 19 | anbi12i 678 | . . . . . . 7 |
21 | eeanv 1913 | . . . . . . 7 | |
22 | eqtr2 2371 | . . . . . . . . . . 11 | |
23 | opth 4602 | . . . . . . . . . . . . . 14 | |
24 | 23 | simplbi 446 | . . . . . . . . . . . . 13 |
25 | 24 | eqcomd 2358 | . . . . . . . . . . . 12 |
26 | 25 | opeq1d 4584 | . . . . . . . . . . 11 |
27 | 22, 26 | syl 15 | . . . . . . . . . 10 |
28 | eqeq1 2359 | . . . . . . . . . . 11 | |
29 | 28 | adantl 452 | . . . . . . . . . 10 |
30 | 27, 29 | mpbird 223 | . . . . . . . . 9 |
31 | 30 | exlimivv 1635 | . . . . . . . 8 |
32 | opeq2 4579 | . . . . . . . . . . . 12 | |
33 | 32 | eqeq2d 2364 | . . . . . . . . . . 11 |
34 | opeq1 4578 | . . . . . . . . . . . 12 | |
35 | 34 | eqeq2d 2364 | . . . . . . . . . . 11 |
36 | 33, 35 | bi2anan9 843 | . . . . . . . . . 10 |
37 | 17, 13, 36 | spc2ev 2947 | . . . . . . . . 9 |
38 | 37 | anidms 626 | . . . . . . . 8 |
39 | 31, 38 | impbii 180 | . . . . . . 7 |
40 | 20, 21, 39 | 3bitr2i 264 | . . . . . 6 |
41 | 40 | anbi2i 675 | . . . . 5 |
42 | 3anass 938 | . . . . 5 | |
43 | 11, 41, 42 | 3bitr4i 268 | . . . 4 |
44 | 10, 43 | bitri 240 | . . 3 |
45 | 44 | 2exbii 1583 | . 2 |
46 | 9, 45 | bitri 240 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wb 176 wa 358 w3a 934 wex 1541 wceq 1642 cin 3208 cop 4561 class class class wbr 4639 c1st 4717 ccom 4721 ccnv 4771 c2nd 4783 ctxp 5735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-13 1712 ax-14 1714 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-pss 3261 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 df-opab 4623 df-br 4640 df-1st 4723 df-co 4726 df-cnv 4785 df-2nd 4797 df-txp 5736 |
This theorem is referenced by: restxp 5786 oqelins4 5794 dmtxp 5802 fntxp 5804 brpprod 5839 |
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