Theorem ecoptocl 3239

Description: Implicit substitution of class for equivalence class of ordered pair.

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 group(s):   x,y,A   x,B,y   x,C,y   x,R,y   ps,x,y

Proof of Theorem ecoptocl

Step	Hyp	Ref	Expression
1		ecoptocl.1	. . 3 |- S = ((B X. C)/.R)
2	1	eleq2i 1153	. 2 |- (A e. S <-> A e. ((B X. C)/.R))
3		elqsi 3228	. . 3 |- (A e. ((B X. C)/.R) -> E.z(z e. (B X. C) /\ A = [z]R))
4		cleqid 1102	. . . . . 6 |- (B X. C) = (B X. C)
5		eceq2 3215	. . . . . . . 8 |- (<.x, y>. = z -> [<.x, y>.]R = [z]R)
6	5	cleq2d 1112	. . . . . . 7 |- (<.x, y>. = z -> (A = [<.x, y>.]R <-> A = [z]R))
7	6	imbi1d 465	. . . . . 6 |- (<.x, y>. = z -> ((A = [<.x, y>.]R -> ps) <-> (A = [z]R -> ps)))
8		ecoptocl.2	. . . . . . . . 9 |- ([<.x, y>.]R = A -> (ph <-> ps))
9	8	cleqcoms 1104	. . . . . . . 8 |- (A = [<.x, y>.]R -> (ph <-> ps))
10		ecoptocl.3	. . . . . . . 8 |- ((x e. B /\ y e. C) -> ph)
11	9, 10	syl5bi 183	. . . . . . 7 |- (A = [<.x, y>.]R -> ((x e. B /\ y e. C) -> ps))
12	11	com12 13	. . . . . 6 |- ((x e. B /\ y e. C) -> (A = [<.x, y>.]R -> ps))
13	4, 7, 12	optocl 2469	. . . . 5 |- (z e. (B X. C) -> (A = [z]R -> ps))
14	13	imp 277	. . . 4 |- ((z e. (B X. C) /\ A = [z]R) -> ps)
15	14	19.23aiv 952	. . 3 |- (E.z(z e. (B X. C) /\ A = [z]R) -> ps)
16	3, 15	syl 12	. 2 |- (A e. ((B X. C)/.R) -> ps)
17	2, 16	sylbi 174	1 |- (A e. S -> ps)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  <.cop 1810   X. cxp 2408  [cec 3198  /.cqs 3199

This theorem is referenced by:  2ecoptocl 3240  3ecoptocl 3241  mulidpq 3863  recmulpq 3864  halfpq 3876  0idsr 4000  1idsr 4001  00sr 4002  recexsrlem 4006  map2psrpr 4014

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-ec 3202  df-qs 3205

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