Theorem iscst4 14522

Description: The value of the couple <.A, B>. through the derived operation H (expressed with a union).

Hypotheses

Ref	Expression
iscst1.1	|- X = dom dom G
iscst1.2	|- H = (cset` G)

Assertion

Ref	Expression
iscst4	|- ((G e. M /\ A e. ~PX /\ B e. ~PX) -> (AHB) = U_x e. B (AH{x}))

Distinct variable groups:   x,A   x,B   x,G   x,M   x,X

Proof of Theorem iscst4

Step	Hyp	Ref	Expression
1		iscst1.1	. . 3 |- X = dom dom G
2		iscst1.2	. . 3 |- H = (cset` G)
3	1, 2	iscst2 14520	. 2 |- ((G e. M /\ A e. ~PX /\ B e. ~PX) -> (AHB) = {a | E.u e. A E.v e. B a = (uGv)})
4		df-rex 2110	. . . . . . . . . 10 |- (E.w e. {x}a = (uGw) <-> E.w(w e. {x} /\ a = (uGw)))
5		elsn 3058	. . . . . . . . . . . 12 |- (w e. {x} <-> w = x)
6	5	anbi1i 539	. . . . . . . . . . 11 |- ((w e. {x} /\ a = (uGw)) <-> (w = x /\ a = (uGw)))
7	6	exbii 1398	. . . . . . . . . 10 |- (E.w(w e. {x} /\ a = (uGw)) <-> E.w(w = x /\ a = (uGw)))
8		visset 2295	. . . . . . . . . . 11 |- x e. _V
9		opreq2 4890	. . . . . . . . . . . 12 |- (w = x -> (uGw) = (uGx))
10	9	eqeq2d 1895	. . . . . . . . . . 11 |- (w = x -> (a = (uGw) <-> a = (uGx)))
11	8, 10	ceqsexv 2325	. . . . . . . . . 10 |- (E.w(w = x /\ a = (uGw)) <-> a = (uGx))
12	4, 7, 11	3bitrri 195	. . . . . . . . 9 |- (a = (uGx) <-> E.w e. {x}a = (uGw))
13	12	rexbii 2128	. . . . . . . 8 |- (E.x e. B a = (uGx) <-> E.x e. B E.w e. {x}a = (uGw))
14	13	a1i 8	. . . . . . 7 |- ((G e. M /\ A e. ~PX /\ B e. ~PX) -> (E.x e. B a = (uGx) <-> E.x e. B E.w e. {x}a = (uGw)))
15		opreq2 4890	. . . . . . . . 9 |- (v = x -> (uGv) = (uGx))
16	15	eqeq2d 1895	. . . . . . . 8 |- (v = x -> (a = (uGv) <-> a = (uGx)))
17	16	cbvrexv 2281	. . . . . . 7 |- (E.v e. B a = (uGv) <-> E.x e. B a = (uGx))
18	14, 17	syl5bb 591	. . . . . 6 |- ((G e. M /\ A e. ~PX /\ B e. ~PX) -> (E.v e. B a = (uGv) <-> E.x e. B E.w e. {x}a = (uGw)))
19	18	rexbidv 2124	. . . . 5 |- ((G e. M /\ A e. ~PX /\ B e. ~PX) -> (E.u e. A E.v e. B a = (uGv) <-> E.u e. A E.x e. B E.w e. {x}a = (uGw)))
20		rexcom 2243	. . . . 5 |- (E.u e. A E.x e. B E.w e. {x}a = (uGw) <-> E.x e. B E.u e. A E.w e. {x}a = (uGw))
21	19, 20	syl6bb 595	. . . 4 |- ((G e. M /\ A e. ~PX /\ B e. ~PX) -> (E.u e. A E.v e. B a = (uGv) <-> E.x e. B E.u e. A E.w e. {x}a = (uGw)))
22	21	abbidv 2008	. . 3 |- ((G e. M /\ A e. ~PX /\ B e. ~PX) -> {a | E.u e. A E.v e. B a = (uGv)} = {a | E.x e. B E.u e. A E.w e. {x}a = (uGw)})
23		iunab 3300	. . 3 |- U_x e. B {a | E.u e. A E.w e. {x}a = (uGw)} = {a | E.x e. B E.u e. A E.w e. {x}a = (uGw)}
24	22, 23	syl6eqr 1946	. 2 |- ((G e. M /\ A e. ~PX /\ B e. ~PX) -> {a | E.u e. A E.v e. B a = (uGv)} = U_x e. B {a | E.u e. A E.w e. {x}a = (uGw)})
25		simpl1 879	. . . . 5 |- (((G e. M /\ A e. ~PX /\ B e. ~PX) /\ x e. B) -> G e. M)
26		simpl2 880	. . . . 5 |- (((G e. M /\ A e. ~PX /\ B e. ~PX) /\ x e. B) -> A e. ~PX)
27		elelpwi 3040	. . . . . . . 8 |- ((x e. B /\ B e. ~PX) -> x e. X)
28	27	ancoms 484	. . . . . . 7 |- ((B e. ~PX /\ x e. B) -> x e. X)
29	8	snelpw 3501	. . . . . . 7 |- (x e. X <-> {x} e. ~PX)
30	28, 29	sylib 215	. . . . . 6 |- ((B e. ~PX /\ x e. B) -> {x} e. ~PX)
31	30	3ad2antl3 1040	. . . . 5 |- (((G e. M /\ A e. ~PX /\ B e. ~PX) /\ x e. B) -> {x} e. ~PX)
32	1, 2	iscst2 14520	. . . . 5 |- ((G e. M /\ A e. ~PX /\ {x} e. ~PX) -> (AH{x}) = {a | E.u e. A E.w e. {x}a = (uGw)})
33	25, 26, 31, 32	syl111anc 1100	. . . 4 |- (((G e. M /\ A e. ~PX /\ B e. ~PX) /\ x e. B) -> (AH{x}) = {a | E.u e. A E.w e. {x}a = (uGw)})
34	33	r19.21aiva 2176	. . 3 |- ((G e. M /\ A e. ~PX /\ B e. ~PX) -> A.x e. B (AH{x}) = {a | E.u e. A E.w e. {x}a = (uGw)})
35		iuneq2 3273	. . . 4 |- (A.x e. B (AH{x}) = {a | E.u e. A E.w e. {x}a = (uGw)} -> U_x e. B (AH{x}) = U_x e. B {a | E.u e. A E.w e. {x}a = (uGw)})
36	35	eqcomd 1889	. . 3 |- (A.x e. B (AH{x}) = {a | E.u e. A E.w e. {x}a = (uGw)} -> U_x e. B {a | E.u e. A E.w e. {x}a = (uGw)} = U_x e. B (AH{x}))
37	34, 36	syl 12	. 2 |- ((G e. M /\ A e. ~PX /\ B e. ~PX) -> U_x e. B {a | E.u e. A E.w e. {x}a = (uGw)} = U_x e. B (AH{x}))
38	3, 24, 37	3eqtrd 1929	1 |- ((G e. M /\ A e. ~PX /\ B e. ~PX) -> (AHB) = U_x e. B (AH{x}))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  {cab 1871  A.wral 2105  E.wrex 2106  ~Pcpw 3032  {csn 3044  U_ciun 3255  dom cdm 3986  ` cfv 3998  (class class class)co 4884  csetccst 14517

This theorem is referenced by:  tpgprop2 14987

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-13 1311  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-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  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-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  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-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-opr 4886  df-oprab 4887  df-cst 14518

Copyright terms: Public domain