Theorem ecoptocl 4303

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 1538	. 2 |- (A e. S <-> A e. ((B X. C)/.R))
3		elqsi 4291	. . 3 |- (A e. ((B X. C)/.R) -> E.z(z e. (B X. C) /\ A = [z]R))
4		eqid 1475	. . . . . 6 |- (B X. C) = (B X. C)
5		eceq2 4278	. . . . . . . 8 |- (<.x, y>. = z -> [<.x, y>.]R = [z]R)
6	5	eqeq2d 1486	. . . . . . 7 |- (<.x, y>. = z -> (A = [<.x, y>.]R <-> A = [z]R))
7	6	imbi1d 613	. . . . . 6 |- (<.x, y>. = z -> ((A = [<.x, y>.]R -> ps) <-> (A = [z]R -> ps)))
8		ecoptocl.2	. . . . . . . 8 |- ([<.x, y>.]R = A -> (ph <-> ps))
9	8	eqcoms 1478	. . . . . . 7 |- (A = [<.x, y>.]R -> (ph <-> ps))
10		ecoptocl.3	. . . . . . 7 |- ((x e. B /\ y e. C) -> ph)
11	9, 10	syl5cbi 209	. . . . . 6 |- ((x e. B /\ y e. C) -> (A = [<.x, y>.]R -> ps))
12	4, 7, 11	optocl 3235	. . . . 5 |- (z e. (B X. C) -> (A = [z]R -> ps))
13	12	imp 350	. . . 4 |- ((z e. (B X. C) /\ A = [z]R) -> ps)
14	13	19.23aiv 1295	. . 3 |- (E.z(z e. (B X. C) /\ A = [z]R) -> ps)
15	3, 14	syl 10	. 2 |- (A e. ((B X. C)/.R) -> ps)
16	2, 15	sylbi 199	1 |- (A e. S -> ps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  <.cop 2411   X. cxp 3168  [cec 4259  /.cqs 4260

This theorem is referenced by:  2ecoptocl 4304  3ecoptocl 4305  mulidpq 5069  recmulpq 5070  halfpq 5082  0idsr 5206  1idsr 5207  00sr 5208  recexsrlem 5212  map2psrpr 5220

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-ec 4263  df-qs 4266

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