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Theorem brtxp 5783

Description: Binary relationship over a tail cross product. (Contributed by SF, 11-Feb-2015.)

**Assertion**
Ref	Expression
brtxp	$/\$ $/\$

Distinct variable groups:

Proof of Theorem brtxp Dummy variables

are mutually distinct and distinct from all other variables.

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-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-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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