Theorem xpassen 3344

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 2488	. . 3 |- (A X. B) e. V
4		xpassen.3	. . 3 |- C e. V
5	3, 4	xpex 2488	. 2 |- ((A X. B) X. C) e. V
6		opex 1893	. . 3 |- <.U.dom {U.dom {x}}, <.U.ran {U.dom {x}}, U.ran {x}>.>. e. V
7	6	a1i 7	. 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 1893	. . 3 |- <.<.U.dom {y}, U.dom {U.ran {y}}>., U.ran {U.ran {y}}>. e. V
9	8	a1i 7	. 2 |- (y e. (A X. (B X. C)) -> <.<.U.dom {y}, U.dom {U.ran {y}}>., U.ran {U.ran {y}}>. e. V)
10		opeq12 1878	. . . . . . . . . . 11 |- ((z = U.dom {U.dom {x}} /\ <.w, v>. = <.U.ran {U.dom {x}}, U.ran {x}>.) -> <.z, <.w, v>.>. = <.U.dom {U.dom {x}}, <.U.ran {U.dom {x}}, U.ran {x}>.>.)
11		sneq 1816	. . . . . . . . . . . . . . . . 17 |- (x = <.<.z, w>., v>. -> {x} = {<.<.z, w>., v>.})
12	11	dmeqd 2533	. . . . . . . . . . . . . . . 16 |- (x = <.<.z, w>., v>. -> dom {x} = dom {<.<.z, w>., v>.})
13	12	unieqd 1929	. . . . . . . . . . . . . . 15 |- (x = <.<.z, w>., v>. -> U.dom {x} = U.dom {<.<.z, w>., v>.})
14	13	sneqd 1818	. . . . . . . . . . . . . 14 |- (x = <.<.z, w>., v>. -> {U.dom {x}} = {U.dom {<.<.z, w>., v>.}})
15	14	dmeqd 2533	. . . . . . . . . . . . 13 |- (x = <.<.z, w>., v>. -> dom {U.dom {x}} = dom {U.dom {<.<.z, w>., v>.}})
16	15	unieqd 1929	. . . . . . . . . . . 12 |- (x = <.<.z, w>., v>. -> U.dom {U.dom {x}} = U.dom {U.dom {<.<.z, w>., v>.}})
17		opex 1893	. . . . . . . . . . . . . . . . 17 |- <.z, w>. e. V
18	17	op1sta 2635	. . . . . . . . . . . . . . . 16 |- U.dom {<.<.z, w>., v>.} = <.z, w>.
19	18	sneqi 1817	. . . . . . . . . . . . . . 15 |- {U.dom {<.<.z, w>., v>.}} = {<.z, w>.}
20	19	dmeqi 2532	. . . . . . . . . . . . . 14 |- dom {U.dom {<.<.z, w>., v>.}} = dom {<.z, w>.}
21	20	unieqi 1928	. . . . . . . . . . . . 13 |- U.dom {U.dom {<.<.z, w>., v>.}} = U.dom {<.z, w>.}
22		visset 1350	. . . . . . . . . . . . . 14 |- z e. V
23	22	op1sta 2635	. . . . . . . . . . . . 13 |- U.dom {<.z, w>.} = z
24	21, 23	eqtr 1119	. . . . . . . . . . . 12 |- U.dom {U.dom {<.<.z, w>., v>.}} = z
25	16, 24	syl6req 1141	. . . . . . . . . . 11 |- (x = <.<.z, w>., v>. -> z = U.dom {U.dom {x}})
26		opeq12 1878	. . . . . . . . . . . 12 |- ((w = U.ran {U.dom {x}} /\ v = U.ran {x}) -> <.w, v>. = <.U.ran {U.dom {x}}, U.ran {x}>.)
27	14	rneqd 2557	. . . . . . . . . . . . . 14 |- (x = <.<.z, w>., v>. -> ran {U.dom {x}} = ran {U.dom {<.<.z, w>., v>.}})
28	27	unieqd 1929	. . . . . . . . . . . . 13 |- (x = <.<.z, w>., v>. -> U.ran {U.dom {x}} = U.ran {U.dom {<.<.z, w>., v>.}})
29	19	rneqi 2556	. . . . . . . . . . . . . . 15 |- ran {U.dom {<.<.z, w>., v>.}} = ran {<.z, w>.}
30	29	unieqi 1928	. . . . . . . . . . . . . 14 |- U.ran {U.dom {<.<.z, w>., v>.}} = U.ran {<.z, w>.}
31		visset 1350	. . . . . . . . . . . . . . 15 |- w e. V
32	22, 31	op2nda 2639	. . . . . . . . . . . . . 14 |- U.ran {<.z, w>.} = w
33	30, 32	eqtr 1119	. . . . . . . . . . . . 13 |- U.ran {U.dom {<.<.z, w>., v>.}} = w
34	28, 33	syl6req 1141	. . . . . . . . . . . 12 |- (x = <.<.z, w>., v>. -> w = U.ran {U.dom {x}})
35	11	rneqd 2557	. . . . . . . . . . . . . 14 |- (x = <.<.z, w>., v>. -> ran {x} = ran {<.<.z, w>., v>.})
36	35	unieqd 1929	. . . . . . . . . . . . 13 |- (x = <.<.z, w>., v>. -> U.ran {x} = U.ran {<.<.z, w>., v>.})
37		visset 1350	. . . . . . . . . . . . . 14 |- v e. V
38	17, 37	op2nda 2639	. . . . . . . . . . . . 13 |- U.ran {<.<.z, w>., v>.} = v
39	36, 38	syl6req 1141	. . . . . . . . . . . 12 |- (x = <.<.z, w>., v>. -> v = U.ran {x})
40	26, 34, 39	sylanc 361	. . . . . . . . . . 11 |- (x = <.<.z, w>., v>. -> <.w, v>. = <.U.ran {U.dom {x}}, U.ran {x}>.)
41	10, 25, 40	sylanc 361	. . . . . . . . . 10 |- (x = <.<.z, w>., v>. -> <.z, <.w, v>.>. = <.U.dom {U.dom {x}}, <.U.ran {U.dom {x}}, U.ran {x}>.>.)
42		opeq12 1878	. . . . . . . . . . 11 |- ((<.z, w>. = <.U.dom {y}, U.dom {U.ran {y}}>. /\ v = U.ran {U.ran {y}}) -> <.<.z, w>., v>. = <.<.U.dom {y}, U.dom {U.ran {y}}>., U.ran {U.ran {y}}>.)
43		opeq12 1878	. . . . . . . . . . . 12 |- ((z = U.dom {y} /\ w = U.dom {U.ran {y}}) -> <.z, w>. = <.U.dom {y}, U.dom {U.ran {y}}>.)
44		sneq 1816	. . . . . . . . . . . . . . 15 |- (y = <.z, <.w, v>.>. -> {y} = {<.z, <.w, v>.>.})
45	44	dmeqd 2533	. . . . . . . . . . . . . 14 |- (y = <.z, <.w, v>.>. -> dom {y} = dom {<.z, <.w, v>.>.})
46	45	unieqd 1929	. . . . . . . . . . . . 13 |- (y = <.z, <.w, v>.>. -> U.dom {y} = U.dom {<.z, <.w, v>.>.})
47	22	op1sta 2635	. . . . . . . . . . . . 13 |- U.dom {<.z, <.w, v>.>.} = z
48	46, 47	syl6req 1141	. . . . . . . . . . . 12 |- (y = <.z, <.w, v>.>. -> z = U.dom {y})
49	44	rneqd 2557	. . . . . . . . . . . . . . . . 17 |- (y = <.z, <.w, v>.>. -> ran {y} = ran {<.z, <.w, v>.>.})
50	49	unieqd 1929	. . . . . . . . . . . . . . . 16 |- (y = <.z, <.w, v>.>. -> U.ran {y} = U.ran {<.z, <.w, v>.>.})
51	50	sneqd 1818	. . . . . . . . . . . . . . 15 |- (y = <.z, <.w, v>.>. -> {U.ran {y}} = {U.ran {<.z, <.w, v>.>.}})
52	51	dmeqd 2533	. . . . . . . . . . . . . 14 |- (y = <.z, <.w, v>.>. -> dom {U.ran {y}} = dom {U.ran {<.z, <.w, v>.>.}})
53	52	unieqd 1929	. . . . . . . . . . . . 13 |- (y = <.z, <.w, v>.>. -> U.dom {U.ran {y}} = U.dom {U.ran {<.z, <.w, v>.>.}})
54		opex 1893	. . . . . . . . . . . . . . . . . 18 |- <.w, v>. e. V
55	22, 54	op2nda 2639	. . . . . . . . . . . . . . . . 17 |- U.ran {<.z, <.w, v>.>.} = <.w, v>.
56	55	sneqi 1817	. . . . . . . . . . . . . . . 16 |- {U.ran {<.z, <.w, v>.>.}} = {<.w, v>.}
57	56	dmeqi 2532	. . . . . . . . . . . . . . 15 |- dom {U.ran {<.z, <.w, v>.>.}} = dom {<.w, v>.}
58	57	unieqi 1928	. . . . . . . . . . . . . 14 |- U.dom {U.ran {<.z, <.w, v>.>.}} = U.dom {<.w, v>.}
59	31	op1sta 2635	. . . . . . . . . . . . . 14 |- U.dom {<.w, v>.} = w
60	58, 59	eqtr 1119	. . . . . . . . . . . . 13 |- U.dom {U.ran {<.z, <.w, v>.>.}} = w
61	53, 60	syl6req 1141	. . . . . . . . . . . 12 |- (y = <.z, <.w, v>.>. -> w = U.dom {U.ran {y}})
62	43, 48, 61	sylanc 361	. . . . . . . . . . 11 |- (y = <.z, <.w, v>.>. -> <.z, w>. = <.U.dom {y}, U.dom {U.ran {y}}>.)
63	51	rneqd 2557	. . . . . . . . . . . . 13 |- (y = <.z, <.w, v>.>. -> ran {U.ran {y}} = ran {U.ran {<.z, <.w, v>.>.}})
64	63	unieqd 1929	. . . . . . . . . . . 12 |- (y = <.z, <.w, v>.>. -> U.ran {U.ran {y}} = U.ran {U.ran {<.z, <.w, v>.>.}})
65	56	rneqi 2556	. . . . . . . . . . . . . 14 |- ran {U.ran {<.z, <.w, v>.>.}} = ran {<.w, v>.}
66	65	unieqi 1928	. . . . . . . . . . . . 13 |- U.ran {U.ran {<.z, <.w, v>.>.}} = U.ran {<.w, v>.}
67	31, 37	op2nda 2639	. . . . . . . . . . . . 13 |- U.ran {<.w, v>.} = v
68	66, 67	eqtr 1119	. . . . . . . . . . . 12 |- U.ran {U.ran {<.z, <.w, v>.>.}} = v
69	64, 68	syl6req 1141	. . . . . . . . . . 11 |- (y = <.z, <.w, v>.>. -> v = U.ran {U.ran {y}})
70	42, 62, 69	sylanc 361	. . . . . . . . . 10