Theorem ecoptocl 5362

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 groups:   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 1961	. 2 |- (A e. S <-> A e. ((B X. C)/.R))
3		elqsi 5349	. . 3 |- (A e. ((B X. C)/.R) -> E.z e. (B X. C)A = [z]R)
4		eqid 1884	. . . . 5 |- (B X. C) = (B X. C)
5		eceq2 5336	. . . . . . 7 |- (<.x, y>. = z -> [<.x, y>.]R = [z]R)
6	5	eqeq2d 1895	. . . . . 6 |- (<.x, y>. = z -> (A = [<.x, y>.]R <-> A = [z]R))
7	6	imbi1d 675	. . . . 5 |- (<.x, y>. = z -> ((A = [<.x, y>.]R -> ps) <-> (A = [z]R -> ps)))
8		ecoptocl.2	. . . . . . 7 |- ([<.x, y>.]R = A -> (ph <-> ps))
9	8	eqcoms 1887	. . . . . 6 |- (A = [<.x, y>.]R -> (ph <-> ps))
10		ecoptocl.3	. . . . . 6 |- ((x e. B /\ y e. C) -> ph)
11	9, 10	syl5cbi 226	. . . . 5 |- ((x e. B /\ y e. C) -> (A = [<.x, y>.]R -> ps))
12	4, 7, 11	optocl 4061	. . . 4 |- (z e. (B X. C) -> (A = [z]R -> ps))
13	12	r19.23aiv 2211	. . 3 |- (E.z e. (B X. C)A = [z]R -> ps)
14	3, 13	syl 12	. 2 |- (A e. ((B X. C)/.R) -> ps)
15	2, 14	sylbi 216	1 |- (A e. S -> ps)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106  <.cop 3046   X. cxp 3984  [cec 5316  /.cqs 5317

This theorem is referenced by:  2ecoptocl 5363  3ecoptocl 5364  mulidpq 6221  recmulpq 6222  halfpq 6234  0idsr 6358  1idsr 6359  00sr 6360  recexsrlem 6364  map2psrpr 6372

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-ec 5320  df-qs 5323

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