Theorem brecop2 7424

Description: Binary relation on a quotient set. Lemma for real number construction. Eliminates antecedent from last hypothesis. (Contributed by NM, 13-Feb-1996.)

Hypotheses

Ref	Expression
brecop2.1
brecop2.5
brecop2.6
brecop2.7
brecop2.8
brecop2.9
brecop2.10
brecop2.11

Assertion

Ref	Expression
brecop2

Proof of Theorem brecop2

Step	Hyp	Ref	Expression
1		brecop2.7	. . . 4
2	1	brel 5053	. . 3
3		brecop2.5	. . . . . . 7
4		ecelqsdm 7400	. . . . . . 7
5	3, 4	mpan 670	. . . . . 6
6		brecop2.6	. . . . . 6
7	5, 6	eleq2s 2565	. . . . 5
8		opelxp 5034	. . . . 5
9	7, 8	sylib 196	. . . 4
10		ecelqsdm 7400	. . . . . . 7
11	3, 10	mpan 670	. . . . . 6
12	11, 6	eleq2s 2565	. . . . 5
13		opelxp 5034	. . . . 5
14	12, 13	sylib 196	. . . 4
15	9, 14	anim12i 566	. . 3
16	2, 15	syl 16	. 2
17		brecop2.8	. . . . 5
18	17	brel 5053	. . . 4
19		brecop2.10	. . . . . 6
20		brecop2.9	. . . . . 6
21	19, 20	ndmovrcl 6461	. . . . 5
22	19, 20	ndmovrcl 6461	. . . . 5
23	21, 22	anim12i 566	. . . 4
24	18, 23	syl 16	. . 3
25		an42 825	. . 3
26	24, 25	sylib 196	. 2
27		brecop2.11	. 2
28	16, 26, 27	pm5.21nii 353	1

Syntax hints:  -.wn 3  ->wi 4  <->wb 184  /\wa 369  =wceq 1395  e.wcel 1818  cvv 3109  C_wss 3475  c0 3784  <.cop 4035   class class class wbr 4452  X.cxp 5002  domcdm 5004  (class class class)co 6296  [cec 7328  /.cqs 7329

This theorem is referenced by:  ltsrpr  9475

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

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-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-xp 5010  df-cnv 5012  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fv 5601  df-ov 6299  df-ec 7332  df-qs 7336