Theorem dualded2lem 15130

Description: Lemma for dualded 15132.

Hypotheses

Ref	Expression
dualded2lem.1	|- D = (dom` T)
dualded2lem.2	|- C = (cod` T)
dualded2lem.3	|- R = (o` T)

Assertion

Ref	Expression
dualded2lem	|- ((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ (D` G) = (C` F)) -> (F{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}G) = (GRF))

Distinct variable groups:   x,D,y,z   x,F,y,z   x,G,y,z   x,R,y,z   x,T,y,z

Proof of Theorem dualded2lem

Step	Hyp	Ref	Expression
1		dedalg 15090	. . . . . . . 8 |- (T e. Ded -> T e. Alg )
2		eqid 1884	. . . . . . . . 9 |- dom D = dom D
3		dualded2lem.1	. . . . . . . . 9 |- D = (dom` T)
4		dualded2lem.3	. . . . . . . . 9 |- R = (o` T)
5	2, 3, 4	cmppfa 15079	. . . . . . . 8 |- (T e. Alg -> (Fun R /\ dom R C_ (dom D X. dom D) /\ ran R C_ dom D))
6	1, 5	syl 12	. . . . . . 7 |- (T e. Ded -> (Fun R /\ dom R C_ (dom D X. dom D) /\ ran R C_ dom D))
7	6	simp2d 889	. . . . . 6 |- (T e. Ded -> dom R C_ (dom D X. dom D))
8	7	sseld 2619	. . . . 5 |- (T e. Ded -> (<.y, x>. e. dom R -> <.y, x>. e. (dom D X. dom D)))
9	8	anim1d 619	. . . 4 |- (T e. Ded -> ((<.y, x>. e. dom R /\ z = (yRx)) -> (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))))
10	9	ssoprab2g 14333	. . 3 |- (T e. Ded -> {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))})
11		cnvxp 4332	. . . . . . . . 9 |- `'(dom D X. dom D) = (dom D X. dom D)
12	11	eqcomi 1888	. . . . . . . 8 |- (dom D X. dom D) = `'(dom D X. dom D)
13	12	eleq2i 1961	. . . . . . 7 |- (<.y, x>. e. (dom D X. dom D) <-> <.y, x>. e. `'(dom D X. dom D))
14		visset 2295	. . . . . . . 8 |- y e. _V
15		visset 2295	. . . . . . . 8 |- x e. _V
16	14, 15	opelcnv 4143	. . . . . . 7 |- (<.y, x>. e. `'(dom D X. dom D) <-> <.x, y>. e. (dom D X. dom D))
17	13, 16	bitri 190	. . . . . 6 |- (<.y, x>. e. (dom D X. dom D) <-> <.x, y>. e. (dom D X. dom D))
18	17	anbi1i 539	. . . . 5 |- ((<.y, x>. e. (dom D X. dom D) /\ z = (yRx)) <-> (<.x, y>. e. (dom D X. dom D) /\ z = (yRx)))
19	18	oprabbii 4923	. . . 4 |- {<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))} = {<.<.x, y>., z>. | (<.x, y>. e. (dom D X. dom D) /\ z = (yRx))}
20		twsvbdop 14332	. . . 4 |- {<.<.x, y>., z>. | (<.x, y>. e. (dom D X. dom D) /\ z = (yRx))} = {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}
21		eqtr 1904	. . . . 5 |- (({<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))} = {<.<.x, y>., z>. | (<.x, y>. e. (dom D X. dom D) /\ z = (yRx))} /\ {<.<.x, y>., z>. | (<.x, y>. e. (dom D X. dom D) /\ z = (yRx))} = {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) -> {<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))} = {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))})
22		sseq2 2639	. . . . . 6 |- ({<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))} = {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))} -> ({<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))} <-> {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}))
23		oprex 4907	. . . . . . . . . . . . 13 |- (yRx) e. _V
24		eqid 1884	. . . . . . . . . . . . 13 |- {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))} = {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}
25	23, 24	fnoprab2 5064	. . . . . . . . . . . 12 |- {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))} Fn (dom D X. dom D)
26		fnfun 4510	. . . . . . . . . . . 12 |- ({<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))} Fn (dom D X. dom D) -> Fun {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))})
27	25, 26	ax-mp 7	. . . . . . . . . . 11 |- Fun {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}
28	27	a1i 8	. . . . . . . . . 10 |- (((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) /\ (D` G) = (C` F)) -> Fun {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))})
29		simpl3 881	. . . . . . . . . 10 |- (((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) /\ (D` G) = (C` F)) -> {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))})
30		dualded2lem.2	. . . . . . . . . . . . . . . 16 |- C = (cod` T)
31	2, 3, 30, 4	cmppfd 15092	. . . . . . . . . . . . . . 15 |- ((T e. Ded /\ F e. dom D /\ G e. dom D) -> (<.G, F>. e. dom R <-> (D` G) = (C` F)))
32	31	3expb 1068	. . . . . . . . . . . . . 14 |- ((T e. Ded /\ (F e. dom D /\ G e. dom D)) -> (<.G, F>. e. dom R <-> (D` G) = (C` F)))
33	32	3adant3 896	. . . . . . . . . . . . 13 |- ((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) -> (<.G, F>. e. dom R <-> (D` G) = (C` F)))
34	33	biimpar 461	. . . . . . . . . . . 12 |- (((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) /\ (D` G) = (C` F)) -> <.G, F>. e. dom R)
35		elisset 2299	. . . . . . . . . . . . . . . 16 |- (F e. dom D -> F e. _V)
36		elisset 2299	. . . . . . . . . . . . . . . 16 |- (G e. dom D -> G e. _V)
37	35, 36	anim12i 360	. . . . . . . . . . . . . . 15 |- ((F e. dom D /\ G e. dom D) -> (F e. _V /\ G e. _V))
38	37	3ad2ant2 898	. . . . . . . . . . . . . 14 |- ((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) -> (F e. _V /\ G e. _V))
39	38	adantr 425	. . . . . . . . . . . . 13 |- (((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) /\ (D` G) = (C` F)) -> (F e. _V /\ G e. _V))
40		opelcnvg 4140	. . . . . . . . . . . . 13 |- ((F e. _V /\ G e. _V) -> (<.F, G>. e. `'dom R <-> <.G, F>. e. dom R))
41	39, 40	syl 12	. . . . . . . . . . . 12 |- (((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) /\ (D` G) = (C` F)) -> (<.F, G>. e. `'dom R <-> <.G, F>. e. dom R))
42	34, 41	mpbird 213	. . . . . . . . . . 11 |- (((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) /\ (D` G) = (C` F)) -> <.F, G>. e. `'dom R)
43		relcnv 4301	. . . . . . . . . . . . . . . 16 |- Rel `'dom R
44	43	a1i 8	. . . . . . . . . . . . . . 15 |- ((T e. Ded /\ (F e. dom D /\ G e. dom D)) -> Rel `'dom R)
45	23	dmoprabss6 14336	. . . . . . . . . . . . . . 15 |- (Rel `'dom R -> dom {<.<.x, y>., z>. | (<.x, y>. e. `'dom R /\ z = (yRx))} = `'dom R)
46	44, 45	syl 12	. . . . . . . . . . . . . 14 |- ((T e. Ded /\ (F e. dom D /\ G e. dom D)) -> dom {<.<.x, y>., z>. | (<.x, y>. e. `'dom R /\ z = (yRx))} = `'dom R)
47	15, 14	opelcnv 4143	. . . . . . . . . . . . . . . . . 18 |- (<.x, y>. e. `'dom R <-> <.y, x>. e. dom R)
48	47	bicomi 189	. . . . . . . . . . . . . . . . 17 |- (<.y, x>. e. dom R <-> <.x, y>. e. `'dom R)
49	48	anbi1i 539	. . . . . . . . . . . . . . . 16 |- ((<.y, x>. e. dom R /\ z = (yRx)) <-> (<.x, y>. e. `'dom R /\ z = (yRx)))
50	49	oprabbii 4923	. . . . . . . . . . . . . . 15 |- {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} = {<.<.x, y>., z>. | (<.x, y>. e. `'dom R /\ z = (yRx))}
51	50	dmeqi 4158	. . . . . . . . . . . . . 14 |- dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} = dom {<.<.x, y>., z>. | (<.x, y>. e. `'dom R /\ z = (yRx))}
52	46, 51	syl5eq 1940	. . . . . . . . . . . . 13 |- ((T e. Ded /\ (F e. dom D /\ G e. dom D)) -> dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} = `'dom R)
53	52	3adant3 896	. . . . . . . . . . . 12 |- ((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) -> dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} = `'dom R)
54	53	adantr 425	. . . . . . . . . . 11 |- (((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) /\ (D` G) = (C` F)) -> dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} = `'dom R)
55	42, 54	eleqtrrd 1974	. . . . . . . . . 10 |- (((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) /\ (D` G) = (C` F)) -> <.F, G>. e. dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))})
56		oprssoprvg 14335	. . . . . . . . . . 11 |- ((Fun {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))} /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))} /\ <.F, G>. e. dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}) -> (F{<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}G) = (F{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}G))
57	56	eqcomd 1889	. . . . . . . . . 10 |- ((Fun {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))} /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))} /\ <.F, G>. e. dom {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}) -> (F{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}G) = (F{<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}G))
58	28, 29, 55, 57	syl111anc 1100	. . . . . . . . 9 |- (((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) /\ (D` G) = (C` F)) -> (F{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}G) = (F{<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}G))
59		simpl2l 929	. . . . . . . . . 10 |- (((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) /\ (D` G) = (C` F)) -> F e. dom D)
60		simpl2r 930	. . . . . . . . . 10 |- (((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) /\ (D` G) = (C` F)) -> G e. dom D)
61		oprex 4907	. . . . . . . . . . 11 |- (GRF) e. _V
62	61	a1i 8	. . . . . . . . . 10 |- (((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) /\ (D` G) = (C` F)) -> (GRF) e. _V)
63		opreq2 4890	. . . . . . . . . . 11 |- (x = F -> (yRx) = (yRF))
64		opreq1 4889	. . . . . . . . . . 11 |- (y = G -> (yRF) = (GRF))
65	63, 64, 24	oprabval2g 4956	. . . . . . . . . 10 |- ((F e. dom D /\ G e. dom D /\ (GRF) e. _V) -> (F{<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}G) = (GRF))
66	59, 60, 62, 65	syl111anc 1100	. . . . . . . . 9 |- (((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) /\ (D` G) = (C` F)) -> (F{<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}G) = (GRF))
67	58, 66	eqtrd 1925	. . . . . . . 8 |- (((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ {<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) /\ (D` G) = (C` F)) -> (F{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}G) = (GRF))
68	67	3exp1 1084	. . . . . . 7 |- (T e. Ded -> ((F e. dom D /\ G e. dom D) -> ({<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))} -> ((D` G) = (C` F) -> (F{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}G) = (GRF)))))
69	68	com3r 39	. . . . . 6 |- ({<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))} -> (T e. Ded -> ((F e. dom D /\ G e. dom D) -> ((D` G) = (C` F) -> (F{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}G) = (GRF)))))
70	22, 69	syl6bi 231	. . . . 5 |- ({<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))} = {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))} -> ({<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))} -> (T e. Ded -> ((F e. dom D /\ G e. dom D) -> ((D` G) = (C` F) -> (F{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}G) = (GRF))))))
71	21, 70	syl 12	. . . 4 |- (({<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))} = {<.<.x, y>., z>. | (<.x, y>. e. (dom D X. dom D) /\ z = (yRx))} /\ {<.<.x, y>., z>. | (<.x, y>. e. (dom D X. dom D) /\ z = (yRx))} = {<.<.x, y>., z>. | ((x e. dom D /\ y e. dom D) /\ z = (yRx))}) -> ({<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))} -> (T e. Ded -> ((F e. dom D /\ G e. dom D) -> ((D` G) = (C` F) -> (F{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}G) = (GRF))))))
72	19, 20, 71	mp2an 761	. . 3 |- ({<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))} C_ {<.<.x, y>., z>. | (<.y, x>. e. (dom D X. dom D) /\ z = (yRx))} -> (T e. Ded -> ((F e. dom D /\ G e. dom D) -> ((D` G) = (C` F) -> (F{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}G) = (GRF)))))
73	10, 72	mpcom 60	. 2 |- (T e. Ded -> ((F e. dom D /\ G e. dom D) -> ((D` G) = (C` F) -> (F{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}G) = (GRF))))
74	73	3imp 1061	1 |- ((T e. Ded /\ (F e. dom D /\ G e. dom D) /\ (D` G) = (C` F)) -> (F{<.<.x, y>., z>. | (<.y, x>. e. dom R /\ z = (yRx))}G) = (GRF))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  _Vcvv 2292   C_ wss 2593  <.cop 3046   X. cxp 3984  `'ccnv 3985  dom cdm 3986  ran crn 3987  Rel wrel 3991  Fun wfun 3992   Fn wfn 3993  ` cfv 3998  (class class class)co 4884  {copab2 4885   Alg calg 15058  domcdom_ 15059  codccod_ 15060  oco_ 15062   Ded cded 15081

This theorem is referenced by:  dualded 15132

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-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-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-int 3215  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-fo 4012  df-fv 4014  df-opr 4886  df-oprab 4887  df-1st 5020  df-2nd 5021  df-alg 15063  df-doma 15064  df-coda 15065  df-ida 15066  df-cmpa 15067  df-ded 15082

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