Theorem tposoprab 7010

Description: Transposition of a class of ordered triples. (Contributed by Mario Carneiro, 10-Sep-2015.)

Hypothesis

Ref	Expression
tposoprab.1

Assertion

Ref	Expression
tposoprab

Distinct variable group:   ,,

Proof of Theorem tposoprab

Dummy variables are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		tposoprab.1	. . 3
2	1	tposeqi 7007	. 2
3		reldmoprab 6387	. . 3
4		dftpos3 6992	. . 3
5	3, 4	ax-mp 5	. 2
6		nfcv 2619	. . . . 5
7		nfoprab2 6347	. . . . 5
8		nfcv 2619	. . . . 5
9	6, 7, 8	nfbr 4496	. . . 4
10		nfcv 2619	. . . . 5
11		nfoprab1 6346	. . . . 5
12		nfcv 2619	. . . . 5
13	10, 11, 12	nfbr 4496	. . . 4
14		nfv 1707	. . . 4
15		nfv 1707	. . . 4
16		opeq12 4219	. . . . . 6
17	16	ancoms 453	. . . . 5
18	17	breq1d 4462	. . . 4
19	9, 13, 14, 15, 18	cbvoprab12 6371	. . 3
20		nfcv 2619	. . . . 5
21		nfoprab3 6348	. . . . 5
22		nfcv 2619	. . . . 5
23	20, 21, 22	nfbr 4496	. . . 4
24		nfv 1707	. . . 4
25		breq2 4456	. . . . 5
26		df-br 4453	. . . . . 6
27		oprabid 6323	. . . . . 6
28	26, 27	bitri 249	. . . . 5
29	25, 28	syl6bb 261	. . . 4
30	23, 24, 29	cbvoprab3 6373	. . 3
31	19, 30	eqtri 2486	. 2
32	2, 5, 31	3eqtri 2490	1

Syntax hints:  /\wa 369  =wceq 1395  e.wcel 1818  <.cop 4035   class class class wbr 4452  domcdm 5004  Relwrel 5009  {coprab 6297  tposctpos 6973

This theorem is referenced by:  tposmpt2  7011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fn 5596  df-fv 5601  df-oprab 6300  df-tpos 6974