Theorem 1ded 15085

Description: Category 1 is a deductive system. We can think of the morphism of Category 1 as corresponding to ph|- ph.

Hypothesis

Ref	Expression
1ded.1	|- A e. _V

Assertion

Ref	Expression
1ded	|- <.<.{<.<.A, A>., A>.}, {<.<.A, A>., A>.}>., <.{<.A, <.A, A>.>.}, {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}>.>. e. Ded

Proof of Theorem 1ded

Step	Hyp	Ref	Expression
1		snex 3492	. . . 4 |- {<.<.A, A>., A>.} e. _V
2		snex 3492	. . . 4 |- {<.A, <.A, A>.>.} e. _V
3	1, 1, 2	3pm3.2i 1048	. . 3 |- ({<.<.A, A>., A>.} e. _V /\ {<.<.A, A>., A>.} e. _V /\ {<.A, <.A, A>.>.} e. _V)
4		snex 3492	. . 3 |- {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} e. _V
5		dmsnop 4367	. . . . 5 |- dom {<.<.A, A>., A>.} = {<.A, A>.}
6	5	eqcomi 1888	. . . 4 |- {<.A, A>.} = dom {<.<.A, A>., A>.}
7		dmsnop 4367	. . . . 5 |- dom {<.A, <.A, A>.>.} = {A}
8	7	eqcomi 1888	. . . 4 |- {A} = dom {<.A, <.A, A>.>.}
9	6, 8	isded 15083	. . 3 |- ((({<.<.A, A>., A>.} e. _V /\ {<.<.A, A>., A>.} e. _V /\ {<.A, <.A, A>.>.} e. _V) /\ {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} e. _V) -> (<.<.{<.<.A, A>., A>.}, {<.<.A, A>., A>.}>., <.{<.A, <.A, A>.>.}, {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}>.>. e. Ded <-> ((<.<.{<.<.A, A>., A>.}, {<.<.A, A>., A>.}>., <.{<.A, <.A, A>.>.}, {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}>.>. e. Alg /\ A.z e. {A} (({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = z /\ ({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = z) /\ A.x e. {<.A, A>.}A.y e. {<.A, A>.} (<.y, x>. e. dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} <-> ({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x))) /\ (A.x e. {<.A, A>.}A.y e. {<.A, A>.} (({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x) -> ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)) = ({<.<.A, A>., A>.}` x)) /\ A.x e. {<.A, A>.}A.y e. {<.A, A>.} (({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x) -> ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)) = ({<.<.A, A>., A>.}` y))))))
10	3, 4, 9	mp2an 761	. 2 |- (<.<.{<.<.A, A>., A>.}, {<.<.A, A>., A>.}>., <.{<.A, <.A, A>.>.}, {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}>.>. e. Ded <-> ((<.<.{<.<.A, A>., A>.}, {<.<.A, A>., A>.}>., <.{<.A, <.A, A>.>.}, {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}>.>. e. Alg /\ A.z e. {A} (({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = z /\ ({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = z) /\ A.x e. {<.A, A>.}A.y e. {<.A, A>.} (<.y, x>. e. dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} <-> ({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x))) /\ (A.x e. {<.A, A>.}A.y e. {<.A, A>.} (({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x) -> ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)) = ({<.<.A, A>., A>.}` x)) /\ A.x e. {<.A, A>.}A.y e. {<.A, A>.} (({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x) -> ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)) = ({<.<.A, A>., A>.}` y)))))
11		1ded.1	. . . 4 |- A e. _V
12	11	1alg 15069	. . 3 |- <.<.{<.<.A, A>., A>.}, {<.<.A, A>., A>.}>., <.{<.A, <.A, A>.>.}, {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}>.>. e. Alg
13		elsni 3066	. . . . . 6 |- (z e. {A} -> z = A)
14		opex 3527	. . . . . . . . . 10 |- <.A, A>. e. _V
15	11, 14	fvsn 4758	. . . . . . . . 9 |- ({<.A, <.A, A>.>.}` A) = <.A, A>.
16	15	fveq2i 4684	. . . . . . . 8 |- ({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` A)) = ({<.<.A, A>., A>.}` <.A, A>.)
17	14, 11	fvsn 4758	. . . . . . . 8 |- ({<.<.A, A>., A>.}` <.A, A>.) = A
18	16, 17	eqtri 1908	. . . . . . 7 |- ({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` A)) = A
19		fveq2 4681	. . . . . . . 8 |- (z = A -> ({<.A, <.A, A>.>.}` z) = ({<.A, <.A, A>.>.}` A))
20	19	fveq2d 4685	. . . . . . 7 |- (z = A -> ({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = ({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` A)))
21		id 73	. . . . . . 7 |- (z = A -> z = A)
22	18, 20, 21	3eqtr4a 1954	. . . . . 6 |- (z = A -> ({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = z)
23	13, 22	syl 12	. . . . 5 |- (z e. {A} -> ({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = z)
24		anidmdbi 481	. . . . 5 |- ((z e. {A} -> (({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = z /\ ({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = z)) <-> (z e. {A} -> ({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = z))
25	23, 24	mpbir 207	. . . 4 |- (z e. {A} -> (({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = z /\ ({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = z))
26	25	rgen 2159	. . 3 |- A.z e. {A} (({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = z /\ ({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = z)
27		opex 3527	. . . . . . . . . 10 |- <.<.A, A>., <.A, A>.>. e. _V
28	27	snid 3069	. . . . . . . . 9 |- <.<.A, A>., <.A, A>.>. e. {<.<.A, A>., <.A, A>.>.}
29		dmsnop 4367	. . . . . . . . 9 |- dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} = {<.<.A, A>., <.A, A>.>.}
30	28, 29	eleqtrri 1970	. . . . . . . 8 |- <.<.A, A>., <.A, A>.>. e. dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}
31		eqid 1884	. . . . . . . 8 |- ({<.<.A, A>., A>.}` <.A, A>.) = ({<.<.A, A>., A>.}` <.A, A>.)
32	30, 31	2th 786	. . . . . . 7 |- (<.<.A, A>., <.A, A>.>. e. dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} <-> ({<.<.A, A>., A>.}` <.A, A>.) = ({<.<.A, A>., A>.}` <.A, A>.))
33	32	a1i 8	. . . . . 6 |- ((x = <.A, A>. /\ y = <.A, A>.) -> (<.<.A, A>., <.A, A>.>. e. dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} <-> ({<.<.A, A>., A>.}` <.A, A>.) = ({<.<.A, A>., A>.}` <.A, A>.)))
34		opeq12 3160	. . . . . . . . 9 |- ((y = <.A, A>. /\ x = <.A, A>.) -> <.y, x>. = <.<.A, A>., <.A, A>.>.)
35	34	eqcomd 1889	. . . . . . . 8 |- ((y = <.A, A>. /\ x = <.A, A>.) -> <.<.A, A>., <.A, A>.>. = <.y, x>.)
36	35	ancoms 484	. . . . . . 7 |- ((x = <.A, A>. /\ y = <.A, A>.) -> <.<.A, A>., <.A, A>.>. = <.y, x>.)
37	36	eleq1d 1963	. . . . . 6 |- ((x = <.A, A>. /\ y = <.A, A>.) -> (<.<.A, A>., <.A, A>.>. e. dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} <-> <.y, x>. e. dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}))
38		id 73	. . . . . . . . . 10 |- (y = <.A, A>. -> y = <.A, A>.)
39	38	eqcomd 1889	. . . . . . . . 9 |- (y = <.A, A>. -> <.A, A>. = y)
40	39	adantl 424	. . . . . . . 8 |- ((x = <.A, A>. /\ y = <.A, A>.) -> <.A, A>. = y)
41	40	fveq2d 4685	. . . . . . 7 |- ((x = <.A, A>. /\ y = <.A, A>.) -> ({<.<.A, A>., A>.}` <.A, A>.) = ({<.<.A, A>., A>.}` y))
42		id 73	. . . . . . . . . 10 |- (x = <.A, A>. -> x = <.A, A>.)
43	42	eqcomd 1889	. . . . . . . . 9 |- (x = <.A, A>. -> <.A, A>. = x)
44	43	fveq2d 4685	. . . . . . . 8 |- (x = <.A, A>. -> ({<.<.A, A>., A>.}` <.A, A>.) = ({<.<.A, A>., A>.}` x))
45	44	adantr 425	. . . . . . 7 |- ((x = <.A, A>. /\ y = <.A, A>.) -> ({<.<.A, A>., A>.}` <.A, A>.) = ({<.<.A, A>., A>.}` x))
46	41, 45	eqeq12d 1899	. . . . . 6 |- ((x = <.A, A>. /\ y = <.A, A>.) -> (({<.<.A, A>., A>.}` <.A, A>.) = ({<.<.A, A>., A>.}` <.A, A>.) <-> ({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x)))
47	33, 37, 46	3bitr3d 607	. . . . 5 |- ((x = <.A, A>. /\ y = <.A, A>.) -> (<.y, x>. e. dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} <-> ({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x)))
48		elsni 3066	. . . . 5 |- (x e. {<.A, A>.} -> x = <.A, A>.)
49		elsni 3066	. . . . 5 |- (y e. {<.A, A>.} -> y = <.A, A>.)
50	47, 48, 49	syl2an 503	. . . 4 |- ((x e. {<.A, A>.} /\ y e. {<.A, A>.}) -> (<.y, x>. e. dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} <-> ({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x)))
51	50	rgen2a 2160	. . 3 |- A.x e. {<.A, A>.}A.y e. {<.A, A>.} (<.y, x>. e. dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} <-> ({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x))
52	12, 26, 51	3pm3.2i 1048	. 2 |- (<.<.{<.<.A, A>., A>.}, {<.<.A, A>., A>.}>., <.{<.A, <.A, A>.>.}, {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}>.>. e. Alg /\ A.z e. {A} (({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = z /\ ({<.<.A, A>., A>.}` ({<.A, <.A, A>.>.}` z)) = z) /\ A.x e. {<.A, A>.}A.y e. {<.A, A>.} (<.y, x>. e. dom {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.} <-> ({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x)))
53		opreq2 4890	. . . . . . . . . 10 |- (x = <.A, A>. -> (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x) = (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.))
54	53	eqcomd 1889	. . . . . . . . 9 |- (x = <.A, A>. -> (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.) = (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x))
55	54	fveq2d 4685	. . . . . . . 8 |- (x = <.A, A>. -> ({<.<.A, A>., A>.}` (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.)) = ({<.<.A, A>., A>.}` (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)))
56		opreq1 4889	. . . . . . . . . 10 |- (y = <.A, A>. -> (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x) = (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x))
57	56	eqcomd 1889	. . . . . . . . 9 |- (y = <.A, A>. -> (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x) = (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x))
58	57	fveq2d 4685	. . . . . . . 8 |- (y = <.A, A>. -> ({<.<.A, A>., A>.}` (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)) = ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)))
59	55, 58	sylan9eq 1948	. . . . . . 7 |- ((x = <.A, A>. /\ y = <.A, A>.) -> ({<.<.A, A>., A>.}` (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.)) = ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)))
60		fveq2 4681	. . . . . . . . 9 |- (x = <.A, A>. -> ({<.<.A, A>., A>.}` x) = ({<.<.A, A>., A>.}` <.A, A>.))
61	60	eqcomd 1889	. . . . . . . 8 |- (x = <.A, A>. -> ({<.<.A, A>., A>.}` <.A, A>.) = ({<.<.A, A>., A>.}` x))
62	61	adantr 425	. . . . . . 7 |- ((x = <.A, A>. /\ y = <.A, A>.) -> ({<.<.A, A>., A>.}` <.A, A>.) = ({<.<.A, A>., A>.}` x))
63	59, 62	eqeq12d 1899	. . . . . 6 |- ((x = <.A, A>. /\ y = <.A, A>.) -> (({<.<.A, A>., A>.}` (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.)) = ({<.<.A, A>., A>.}` <.A, A>.) <-> ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)) = ({<.<.A, A>., A>.}` x)))
64		df-opr 4886	. . . . . . . . 9 |- (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.) = ({<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}` <.<.A, A>., <.A, A>.>.)
65	27, 14	fvsn 4758	. . . . . . . . 9 |- ({<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}` <.<.A, A>., <.A, A>.>.) = <.A, A>.
66	64, 65	eqtri 1908	. . . . . . . 8 |- (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.) = <.A, A>.
67	66	fveq2i 4684	. . . . . . 7 |- ({<.<.A, A>., A>.}` (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.)) = ({<.<.A, A>., A>.}` <.A, A>.)
68	67	a1i 8	. . . . . 6 |- (({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x) -> ({<.<.A, A>., A>.}` (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.)) = ({<.<.A, A>., A>.}` <.A, A>.))
69	63, 68	syl5bi 225	. . . . 5 |- ((x = <.A, A>. /\ y = <.A, A>.) -> (({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x) -> ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)) = ({<.<.A, A>., A>.}` x)))
70	69, 48, 49	syl2an 503	. . . 4 |- ((x e. {<.A, A>.} /\ y e. {<.A, A>.}) -> (({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x) -> ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)) = ({<.<.A, A>., A>.}` x)))
71	70	rgen2a 2160	. . 3 |- A.x e. {<.A, A>.}A.y e. {<.A, A>.} (({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x) -> ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)) = ({<.<.A, A>., A>.}` x))
72		opreq1 4889	. . . . . . . . . 10 |- (y = <.A, A>. -> (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.) = (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.))
73	72	eqcomd 1889	. . . . . . . . 9 |- (y = <.A, A>. -> (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.) = (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.))
74		opreq2 4890	. . . . . . . . . 10 |- (x = <.A, A>. -> (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x) = (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.))
75	74	eqcomd 1889	. . . . . . . . 9 |- (x = <.A, A>. -> (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.) = (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x))
76	73, 75	sylan9eqr 1951	. . . . . . . 8 |- ((x = <.A, A>. /\ y = <.A, A>.) -> (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.) = (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x))
77	76	fveq2d 4685	. . . . . . 7 |- ((x = <.A, A>. /\ y = <.A, A>.) -> ({<.<.A, A>., A>.}` (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.)) = ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)))
78		fveq2 4681	. . . . . . . . 9 |- (y = <.A, A>. -> ({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` <.A, A>.))
79	78	eqcomd 1889	. . . . . . . 8 |- (y = <.A, A>. -> ({<.<.A, A>., A>.}` <.A, A>.) = ({<.<.A, A>., A>.}` y))
80	79	adantl 424	. . . . . . 7 |- ((x = <.A, A>. /\ y = <.A, A>.) -> ({<.<.A, A>., A>.}` <.A, A>.) = ({<.<.A, A>., A>.}` y))
81	77, 80	eqeq12d 1899	. . . . . 6 |- ((x = <.A, A>. /\ y = <.A, A>.) -> (({<.<.A, A>., A>.}` (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.)) = ({<.<.A, A>., A>.}` <.A, A>.) <-> ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)) = ({<.<.A, A>., A>.}` y)))
82	64	fveq2i 4684	. . . . . . . 8 |- ({<.<.A, A>., A>.}` (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.)) = ({<.<.A, A>., A>.}` ({<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}` <.<.A, A>., <.A, A>.>.))
83	65	fveq2i 4684	. . . . . . . 8 |- ({<.<.A, A>., A>.}` ({<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}` <.<.A, A>., <.A, A>.>.)) = ({<.<.A, A>., A>.}` <.A, A>.)
84	82, 83	eqtri 1908	. . . . . . 7 |- ({<.<.A, A>., A>.}` (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.)) = ({<.<.A, A>., A>.}` <.A, A>.)
85	84	a1i 8	. . . . . 6 |- (({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x) -> ({<.<.A, A>., A>.}` (<.A, A>.{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}<.A, A>.)) = ({<.<.A, A>., A>.}` <.A, A>.))
86	81, 85	syl5bi 225	. . . . 5 |- ((x = <.A, A>. /\ y = <.A, A>.) -> (({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x) -> ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)) = ({<.<.A, A>., A>.}` y)))
87	86, 48, 49	syl2an 503	. . . 4 |- ((x e. {<.A, A>.} /\ y e. {<.A, A>.}) -> (({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x) -> ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)) = ({<.<.A, A>., A>.}` y)))
88	87	rgen2a 2160	. . 3 |- A.x e. {<.A, A>.}A.y e. {<.A, A>.} (({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x) -> ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)) = ({<.<.A, A>., A>.}` y))
89	71, 88	pm3.2i 307	. 2 |- (A.x e. {<.A, A>.}A.y e. {<.A, A>.} (({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x) -> ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)) = ({<.<.A, A>., A>.}` x)) /\ A.x e. {<.A, A>.}A.y e. {<.A, A>.} (({<.<.A, A>., A>.}` y) = ({<.<.A, A>., A>.}` x) -> ({<.<.A, A>., A>.}` (y{<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}x)) = ({<.<.A, A>., A>.}` y)))
90	10, 52, 89	mpbir2an 800	1 |- <.<.{<.<.A, A>., A>.}, {<.<.A, A>., A>.}>., <.{<.A, <.A, A>.>.}, {<.<.<.A, A>., <.A, A>.>., <.A, A>.>.}>.>. e. Ded

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  {csn 3044  <.cop 3046  dom cdm 3986  ` cfv 3998  (class class class)co 4884   Alg calg 15058   Ded cded 15081

This theorem is referenced by:  1cat 15106

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-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-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-uni 3178  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-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-alg 15063  df-ded 15082

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