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Mirrors > Home > NFE Home > Th. List > dfxp2 | Unicode version |
Description: Define cross product via the set construction functions. (Contributed by SF, 8-Jan-2015.) |
Ref | Expression |
---|---|
dfxp2 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eeanv 1913 | . . . . 5 | |
2 | vex 2862 | . . . . . . . 8 | |
3 | vex 2862 | . . . . . . . 8 | |
4 | opeq2 4579 | . . . . . . . . . 10 | |
5 | 4 | eqeq2d 2364 | . . . . . . . . 9 |
6 | opeq1 4578 | . . . . . . . . . 10 | |
7 | 6 | eqeq2d 2364 | . . . . . . . . 9 |
8 | 5, 7 | bi2anan9 843 | . . . . . . . 8 |
9 | 2, 3, 8 | spc2ev 2947 | . . . . . . 7 |
10 | 9 | anidms 626 | . . . . . 6 |
11 | simpl 443 | . . . . . . . 8 | |
12 | eqtr2 2371 | . . . . . . . . 9 | |
13 | opth 4602 | . . . . . . . . . 10 | |
14 | 4 | adantl 452 | . . . . . . . . . 10 |
15 | 13, 14 | sylbi 187 | . . . . . . . . 9 |
16 | 12, 15 | syl 15 | . . . . . . . 8 |
17 | 11, 16 | eqtrd 2385 | . . . . . . 7 |
18 | 17 | exlimivv 1635 | . . . . . 6 |
19 | 10, 18 | impbii 180 | . . . . 5 |
20 | brcnv 4892 | . . . . . . 7 | |
21 | 3 | br1st 4858 | . . . . . . 7 |
22 | 20, 21 | bitri 240 | . . . . . 6 |
23 | brcnv 4892 | . . . . . . 7 | |
24 | 2 | br2nd 4859 | . . . . . . 7 |
25 | 23, 24 | bitri 240 | . . . . . 6 |
26 | 22, 25 | anbi12i 678 | . . . . 5 |
27 | 1, 19, 26 | 3bitr4i 268 | . . . 4 |
28 | 27 | 2rexbii 2641 | . . 3 |
29 | elxp2 4802 | . . 3 | |
30 | elima 4754 | . . . . 5 | |
31 | elima 4754 | . . . . 5 | |
32 | 30, 31 | anbi12i 678 | . . . 4 |
33 | elin 3219 | . . . 4 | |
34 | reeanv 2778 | . . . 4 | |
35 | 32, 33, 34 | 3bitr4i 268 | . . 3 |
36 | 28, 29, 35 | 3bitr4i 268 | . 2 |
37 | 36 | eqriv 2350 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wa 358 wex 1541 wceq 1642 wcel 1710 wrex 2615 cin 3208 cop 4561 class class class wbr 4639 c1st 4717 cima 4722 cxp 4770 ccnv 4771 c2nd 4783 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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-ima 4727 df-xp 4784 df-cnv 4785 df-2nd 4797 |
This theorem is referenced by: xpexg 5114 |
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