Theorem ecoptocl 5997

Description: Implicit substitution of class for equivalence class of ordered pair. (Contributed by set.mm contributors, 23-Jul-1995.)

Hypotheses

Ref	Expression
ecoptocl.1	|- S = ((B X. C)/.R)
ecoptocl.2	|- ([<.x, y>.]R = A -> (ph <-> ps))
ecoptocl.3	|- ((x e. B /\ y e. C) -> ph)

Assertion

Ref	Expression
ecoptocl	|- (A e. S -> ps)

Distinct variable groups:   x,y,A   x,B,y   x,C,y   x,R,y   ps,x,y

Allowed substitution hints:   ph(x,y)   S(x,y)

Proof of Theorem ecoptocl

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elqsi 5979	. . 3 |- (A e. ((B X. C)/.R) -> E.z e. (B X. C)A = [z]R)
2		eqid 2353	. . . . 5 |- (B X. C) = (B X. C)
3		eceq1 5963	. . . . . . 7 |- (<.x, y>. = z -> [<.x, y>.]R = [z]R)
4	3	eqeq2d 2364	. . . . . 6 |- (<.x, y>. = z -> (A = [<.x, y>.]R <-> A = [z]R))
5	4	imbi1d 308	. . . . 5 |- (<.x, y>. = z -> ((A = [<.x, y>.]R -> ps) <-> (A = [z]R -> ps)))
6		ecoptocl.3	. . . . . 6 |- ((x e. B /\ y e. C) -> ph)
7		ecoptocl.2	. . . . . . 7 |- ([<.x, y>.]R = A -> (ph <-> ps))
8	7	eqcoms 2356	. . . . . 6 |- (A = [<.x, y>.]R -> (ph <-> ps))
9	6, 8	syl5ibcom 211	. . . . 5 |- ((x e. B /\ y e. C) -> (A = [<.x, y>.]R -> ps))
10	2, 5, 9	optocl 4839	. . . 4 |- (z e. (B X. C) -> (A = [z]R -> ps))
11	10	rexlimiv 2733	. . 3 |- (E.z e. (B X. C)A = [z]R -> ps)
12	1, 11	syl 15	. 2 |- (A e. ((B X. C)/.R) -> ps)
13		ecoptocl.1	. 2 |- S = ((B X. C)/.R)
14	12, 13	eleq2s 2445	1 |- (A e. S -> ps)

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   = wceq 1642   e. wcel 1710  E.wrex 2616  <.cop 4562   X. cxp 4771  [cec 5946  /.cqs 5947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-ima 4728  df-xp 4785  df-ec 5948  df-qs 5952

This theorem is referenced by:  2ecoptocl  5998  3ecoptocl  5999