Theorem resoprab 6398

Description: Restriction of an operation class abstraction. (Contributed by NM, 10-Feb-2007.)

Assertion

Ref	Expression
resoprab

Distinct variable groups:   ,,,   ,,,

Proof of Theorem resoprab

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		resopab 5325	. . 3
2		19.42vv 1777	. . . . 5
3		an12 797	. . . . . . 7
4		eleq1 2529	. . . . . . . . . 10
5		opelxp 5034	. . . . . . . . . 10
6	4, 5	syl6bb 261	. . . . . . . . 9
7	6	anbi1d 704	. . . . . . . 8
8	7	pm5.32i 637	. . . . . . 7
9	3, 8	bitri 249	. . . . . 6
10	9	2exbii 1668	. . . . 5
11	2, 10	bitr3i 251	. . . 4
12	11	opabbii 4516	. . 3
13	1, 12	eqtri 2486	. 2
14		dfoprab2 6343	. . 3
15	14	reseq1i 5274	. 2
16		dfoprab2 6343	. 2
17	13, 15, 16	3eqtr4i 2496	1

Syntax hints:  /\wa 369  =wceq 1395  E.wex 1612  e.wcel 1818  <.cop 4035  {copab 4509  X.cxp 5002  |`cres 5006  {coprab 6297

This theorem is referenced by:  resoprab2  6399  df1stres  27522  df2ndres  27523

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-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  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-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-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-opab 4511  df-xp 5010  df-rel 5011  df-res 5016  df-oprab 6300