Theorem xpassen 4441

Description: Associative law for equinumerosity of cross product. Proposition 4.22(e) of [Mendelson] p. 254.

Hypotheses

Ref	Expression
xpassen.1	|- A e. V
xpassen.2	|- B e. V
xpassen.3	|- C e. V

Assertion

Ref	Expression
xpassen	|- ((A X. B) X. C) ~~ (A X. (B X. C))

Proof of Theorem xpassen

Step	Hyp	Ref	Expression
1		xpassen.1	. . . 4 |- A e. V
2		xpassen.2	. . . 4 |- B e. V
3	1, 2	xpex 3260	. . 3 |- (A X. B) e. V
4		xpassen.3	. . 3 |- C e. V
5	3, 4	xpex 3260	. 2 |- ((A X. B) X. C) e. V
6		opex 2782	. . 3 |- <.U.dom {U.dom { x}}, <.U.ran {U.dom { x}}, U.ran { x}>.>. e. V
7	6	a1i 8	. 2 |- (x e. ((A X. B) X. C) -> <.U.dom {U.dom { x}}, <.U.ran {U.dom { x}}, U.ran { x}>.>. e. V)
8		opex 2782	. . 3 |- <.<.U.dom { y}, U.dom {U.ran { y}}>., U.ran {U.ran { y}}>. e. V
9	8	a1i 8	. 2 |- (y e. (A X. (B X. C)) -> <.<.U.dom { y}, U.dom {U.ran { y}}>., U.ran {U.ran { y}}>. e. V)
10		sneq 2417	. . . . . . . . . . . . . . . . 17 |- (x = <.<.z, w>., v>. -> {x} = {<.<.z, w>., v>.})
11	10	dmeqd 3313	. . . . . . . . . . . . . . . 16 |- (x = <.<.z, w>., v>. -> dom { x} = dom {<.<.z, w>., v>.})
12	11	unieqd 2512	. . . . . . . . . . . . . . 15 |- (x = <.<.z, w>., v>. -> U.dom { x} = U.dom {<.<.z, w>., v>.})
13	12	sneqd 2419	. . . . . . . . . . . . . 14 |- (x = <.<.z, w>., v>. -> {U.dom { x}} = {U.dom {<.<.z, w>., v>.}})
14	13	dmeqd 3313	. . . . . . . . . . . . 13 |- (x = <.<.z, w>., v>. -> dom {U.dom { x}} = dom {U.dom {<.<.z, w>., v>.}})
15	14	unieqd 2512	. . . . . . . . . . . 12 |- (x = <.<.z, w>., v>. -> U.dom {U.dom { x}} = U.dom {U.dom {<.<.z, w>., v>.}})
16		opex 2782	. . . . . . . . . . . . . . . . 17 |- <.z, w>. e. V
17	16	op1sta 3448	. . . . . . . . . . . . . . . 16 |- U.dom {<.<.z, w>., v>.} = <.z, w>.
18	17	sneqi 2418	. . . . . . . . . . . . . . 15 |- {U.dom {<.<.z, w>., v>.}} = {<.z, w>.}
19	18	dmeqi 3312	. . . . . . . . . . . . . 14 |- dom {U.dom {<.<.z, w>., v>.}} = dom {<.z, w>.}
20	19	unieqi 2511	. . . . . . . . . . . . 13 |- U.dom {U.dom {<.<.z, w>., v>.}} = U.dom {<.z, w>.}
21		visset 1813	. . . . . . . . . . . . . 14 |- z e. V
22	21	op1sta 3448	. . . . . . . . . . . . 13 |- U.dom {<.z, w>.} = z
23	20, 22	eqtr 1495	. . . . . . . . . . . 12 |- U.dom {U.dom {<.<.z, w>., v>.}} = z
24	15, 23	syl6req 1524	. . . . . . . . . . 11 |- (x = <.<.z, w>., v>. -> z = U.dom {U.dom { x}})
25	13	rneqd 3341	. . . . . . . . . . . . . 14 |- (x = <.<.z, w>., v>. -> ran {U.dom { x}} = ran {U.dom {<.<.z, w>., v>.}})
26	25	unieqd 2512	. . . . . . . . . . . . 13 |- (x = <.<.z, w>., v>. -> U.ran {U.dom { x}} = U.ran {U.dom {<.<.z, w>., v>.}})
27	18	rneqi 3340	. . . . . . . . . . . . . . 15 |- ran {U.dom {<.<.z, w>., v>.}} = ran {<.z, w>.}
28	27	unieqi 2511	. . . . . . . . . . . . . 14 |- U.ran {U.dom {<.<.z, w>., v>.}} = U.ran {<.z, w>.}
29		visset 1813	. . . . . . . . . . . . . . 15 |- w e. V
30	21, 29	op2nda 3452	. . . . . . . . . . . . . 14 |- U.ran {<.z, w>.} = w
31	28, 30	eqtr 1495	. . . . . . . . . . . . 13 |- U.ran {U.dom {<.<.z, w>., v>.}} = w
32	26, 31	syl6req 1524	. . . . . . . . . . . 12 |- (x = <.<.z, w>., v>. -> w = U.ran {U.dom { x}})
33	10	rneqd 3341	. . . . . . . . . . . . . 14 |- (x = <.<.z, w>., v>. -> ran { x} = ran {<.<.z, w>., v>.})
34	33	unieqd 2512	. . . . . . . . . . . . 13 |- (x = <.<.z, w>., v>. -> U.ran { x} = U.ran {<.<.z, w>., v>.})
35		visset 1813	. . . . . . . . . . . . . 14 |- v e. V
36	16, 35	op2nda 3452	. . . . . . . . . . . . 13 |- U.ran {<.<.z, w>., v>.} = v
37	34, 36	syl6req 1524	. . . . . . . . . . . 12 |- (x = <.<.z, w>., v>. -> v = U.ran { x})
38	32, 37	opeq12d 2495	. . . . . . . . . . 11 |- (x = <.<.z, w>., v>. -> <.w, v>. = <.U.ran {U.dom { x}}, U.ran { x}>.)
39	24, 38	opeq12d 2495	. . . . . . . . . 10 |- (x = <.<.z, w>., v>. -> <.z, <.w, v>.>. = <.U.dom {U.dom { x}}, <.U.ran {U.dom { x}}, U.ran { x}>.>.)
40		sneq 2417	. . . . . . . . . . . . . . 15 |- (y = <.z, <.w, v>.>. -> {y} = {<.z, <.w, v>.>.})
41	40	dmeqd 3313	. . . . . . . . . . . . . 14 |- (y = <.z, <.w, v>.>. -> dom { y} = dom {<.z, <.w, v>.>.})
42	41	unieqd 2512	. . . . . . . . . . . . 13 |- (y = <.z, <.w, v>.>. -> U.dom { y} = U.dom {<.z, <.w, v>.>.})
43	21	op1sta 3448	. . . . . . . . . . . . 13 |- U.dom {<.z, <.w, v>.>.} = z
44	42, 43	syl6req 1524	. . . . . . . . . . . 12 |- (y = <.z, <.w, v>.>. -> z = U.dom { y})
45	40	rneqd 3341	. . . . . . . . . . . . . . . . 17 |- (y = <.z, <.w, v>.>. -> ran { y} = ran {<.z, <.w, v>.>.})
46	45	unieqd 2512	. . . . . . . . . . . . . . . 16 |- (y = <.z, <.w, v>.>. -> U.ran { y} = U.ran {<.z, <.w, v>.>.})
47	46	sneqd 2419	. . . . . . . . . . . . . . 15 |- (y = <.z, <.w, v>.>. -> {U.ran { y}} = {U.ran {<.z, <.w, v>.>.}})
48	47	dmeqd 3313	. . . . . . . . . . . . . 14 |- (y = <.z, <.w, v>.>. -> dom {U.ran { y}} = dom {U.ran {<.z, <.w, v>.>.}})
49	48	unieqd 2512	. . . . . . . . . . . . 13 |- (y = <.z, <.w, v>.>. -> U.dom {U.ran { y}} = U.dom {U.ran {<.z, <.w, v>.>.}})
50		opex 2782	. . . . . . . . . . . . . . . . . 18 |- <.w, v>. e. V
51	21, 50	op2nda 3452	. . . . . . . . . . . . . . . . 17 |- U.ran {<.z, <.w, v>.>.} = <.w, v>.
52	51	sneqi 2418	. . . . . . . . . . . . . . . 16 |- {U.ran {<.z, <.w, v>.>.}} = {<.w, v>.}
53	52	dmeqi 3312	. . . . . . . . . . . . . . 15 |- dom {U.ran {<.z, <.w, v>.>.}} = dom {<.w, v>.}
54	53	unieqi 2511	. . . . . . . . . . . . . 14 |- U.dom {U.ran {<.z, <.w, v>.>.}} = U.dom {<.w, v>.}
55	29	op1sta 3448	. . . . . . . . . . . . . 14 |- U.dom {<.w, v>.} = w
56	54, 55	eqtr 1495	. . . . . . . . . . . . 13 |- U.dom {U.ran {<.z, <.w, v>.>.}} = w
57	49, 56	syl6req 1524	. . . . . . . . . . . 12 |- (y = <.z, <.w, v>.>. -> w = U.dom {U.ran { y}})
58	44, 57	opeq12d 2495	. . . . . . . . . . 11 |- (y = <.z, <.w, v>.>. -> <.z, w>. = <.U.dom { y}, U.dom {U.ran { y}}>.)
59	47	rneqd 3341	. . . . . . . . . . . . 13 |- (y = <.z, <.w, v>.>. -> ran {U.ran { y}} = ran {U.ran {<.z, <.w, v>.>.}})
60	59	unieqd 2512	. . . . . . . . . . . 12 |- (y = <.z, <.w, v>.>. -> U.ran {U.ran { y}} = U.ran {U.ran {<.z, <.w, v>.>.}})
61	52	rneqi 3340	. . . . . . . . . . . . . 14 |- ran {U.ran {<.z, <.w, v>.>.}} = ran {<.w, v>.}
62	61	unieqi 2511	. . . . . . . . . . . . 13 |- U.ran {U.ran {<.z, <.w, v>.>.}} = U.ran {<.w, v>.}
63	29, 35	op2nda 3452	. . . . . . . . . . . . 13 |- U.ran {<.w, v>.} = v
64	62, 63	eqtr 1495	. . . . . . . . . . . 12 |- U.ran {U.ran {<.z, <.w, v>.>.}} = v
65	60, 64	syl6req 1524	. . . . . . . . . . 11 |- (y = <.z, <.w, v>.>. -> v = U.ran {U.ran { y}})
66	58, 65	opeq12d 2495	. . . . . . . . . 10 |- (y = <.z, <.w, v>.>. -> <.<.z, w>., v>. = <.<.U.dom { y}, U.dom {U.ran { y}}>., U.ran {U.ran { y}}>.)
67	39, 66	eq2tr 1533	. . . . . . . . 9 |- ((x = <.<.z, w>., v>. /\ y = <.U.dom {U.dom { x}}, <.U.ran {U.dom { x}}, U.ran { x}>.>.) <-> (y = <.z, <.w, v>.>. /\ x = <.<.U.dom { y}, U.dom {U.ran { y}}>., U.ran {U.ran { y}}>.))
68		anass 439	. . . . . . . . 9 |- (((z e. A /\ w e. B) /\ v e. C) <-> (z e. A /\ (w e. B /\ v e. C)))
69	67, 68	anbi12i 482	. . . . . . . 8 |- (((x = <.<.z