Theorem fparlem4 5084

Description: Lemma for fpar 5085.

Assertion

Ref	Expression
fparlem4	|- (G Fn B -> (`'(2nd |` (_V X. _V)) o. (G o. (2nd |` (_V X. _V)))) = U_y e. B ((_V X. {y}) X. (_V X. {(G` y)})))

Distinct variable groups:   y,B   y,G

Proof of Theorem fparlem4

Step	Hyp	Ref	Expression
1		inss1 2812	. . . . . . 7 |- (dom G i^i ran (2nd |` (_V X. _V))) C_ dom G
2	1	a1i 8	. . . . . 6 |- (G Fn B -> (dom G i^i ran (2nd |` (_V X. _V))) C_ dom G)
3		fndm 4512	. . . . . 6 |- (G Fn B -> dom G = B)
4	2, 3	sseqtrd 2653	. . . . 5 |- (G Fn B -> (dom G i^i ran (2nd |` (_V X. _V))) C_ B)
5		dfco2a 4394	. . . . 5 |- ((dom G i^i ran (2nd |` (_V X. _V))) C_ B -> (G o. (2nd |` (_V X. _V))) = U_y e. B ((`'(2nd |` (_V X. _V))"{y}) X. (G"{y})))
6	4, 5	syl 12	. . . 4 |- (G Fn B -> (G o. (2nd |` (_V X. _V))) = U_y e. B ((`'(2nd |` (_V X. _V))"{y}) X. (G"{y})))
7		fnsnfv 4728	. . . . . . 7 |- ((G Fn B /\ y e. B) -> {(G` y)} = (G"{y}))
8		xpeq2 4017	. . . . . . 7 |- ({(G` y)} = (G"{y}) -> ((`'(2nd |` (_V X. _V))"{y}) X. {(G` y)}) = ((`'(2nd |` (_V X. _V))"{y}) X. (G"{y})))
9	7, 8	syl 12	. . . . . 6 |- ((G Fn B /\ y e. B) -> ((`'(2nd |` (_V X. _V))"{y}) X. {(G` y)}) = ((`'(2nd |` (_V X. _V))"{y}) X. (G"{y})))
10		fparlem2 5082	. . . . . . 7 |- (`'(2nd |` (_V X. _V))"{y}) = (_V X. {y})
11	10	xpeq1i 4021	. . . . . 6 |- ((`'(2nd |` (_V X. _V))"{y}) X. {(G` y)}) = ((_V X. {y}) X. {(G` y)})
12	9, 11	syl5eqr 1942	. . . . 5 |- ((G Fn B /\ y e. B) -> ((_V X. {y}) X. {(G` y)}) = ((`'(2nd |` (_V X. _V))"{y}) X. (G"{y})))
13	12	iuneq2dv 3279	. . . 4 |- (G Fn B -> U_y e. B ((_V X. {y}) X. {(G` y)}) = U_y e. B ((`'(2nd |` (_V X. _V))"{y}) X. (G"{y})))
14	6, 13	eqtr4d 1928	. . 3 |- (G Fn B -> (G o. (2nd |` (_V X. _V))) = U_y e. B ((_V X. {y}) X. {(G` y)}))
15	14	coeq2d 4128	. 2 |- (G Fn B -> (`'(2nd |` (_V X. _V)) o. (G o. (2nd |` (_V X. _V)))) = (`'(2nd |` (_V X. _V)) o. U_y e. B ((_V X. {y}) X. {(G` y)})))
16		coiun 4407	. . 3 |- (`'(2nd |` (_V X. _V)) o. U_y e. B ((_V X. {y}) X. {(G` y)})) = U_y e. B (`'(2nd |` (_V X. _V)) o. ((_V X. {y}) X. {(G` y)}))
17	16	a1i 8	. 2 |- (G Fn B -> (`'(2nd |` (_V X. _V)) o. U_y e. B ((_V X. {y}) X. {(G` y)})) = U_y e. B (`'(2nd |` (_V X. _V)) o. ((_V X. {y}) X. {(G` y)})))
18		cnvco 4145	. . . . . 6 |- `'(({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) = (`'(2nd |` (_V X. _V)) o. `'({(G` y)} X. (_V X. {y})))
19		cnvxp 4332	. . . . . . 7 |- `'({(G` y)} X. (_V X. {y})) = ((_V X. {y}) X. {(G` y)})
20	19	coeq2i 4126	. . . . . 6 |- (`'(2nd |` (_V X. _V)) o. `'({(G` y)} X. (_V X. {y}))) = (`'(2nd |` (_V X. _V)) o. ((_V X. {y}) X. {(G` y)}))
21	18, 20	eqtr2i 1909	. . . . 5 |- (`'(2nd |` (_V X. _V)) o. ((_V X. {y}) X. {(G` y)})) = `'(({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V)))
22		visset 2295	. . . . . . . . . . . . 13 |- z e. _V
23	22	opelxp 4036	. . . . . . . . . . . 12 |- (<.x, z>. e. (_V X. {(G` y)}) <-> (x e. _V /\ z e. {(G` y)}))
24		visset 2295	. . . . . . . . . . . . 13 |- x e. _V
25	24	biantrur 794	. . . . . . . . . . . 12 |- (z e. {(G` y)} <-> (x e. _V /\ z e. {(G` y)}))
26		elsn 3058	. . . . . . . . . . . 12 |- (z e. {(G` y)} <-> z = (G` y))
27	23, 25, 26	3bitr2i 196	. . . . . . . . . . 11 |- (<.x, z>. e. (_V X. {(G` y)}) <-> z = (G` y))
28	27	anbi1i 539	. . . . . . . . . 10 |- ((<.x, z>. e. (_V X. {(G` y)}) /\ w e. (_V X. {y})) <-> (z = (G` y) /\ w e. (_V X. {y})))
29		visset 2295	. . . . . . . . . . 11 |- w e. _V
30	29	opelxp 4036	. . . . . . . . . 10 |- (<.<.x, z>., w>. e. ((_V X. {(G` y)}) X. (_V X. {y})) <-> (<.x, z>. e. (_V X. {(G` y)}) /\ w e. (_V X. {y})))
31		opex 3527	. . . . . . . . . . . 12 |- <.x, z>. e. _V
32	31, 29	opelco 4130	. . . . . . . . . . 11 |- (<.<.x, z>., w>. e. (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) <-> E.v(<.x, z>.(2nd |` (_V X. _V))v /\ v({(G` y)} X. (_V X. {y}))w))
33		df-br 3339	. . . . . . . . . . . . . 14 |- (<.x, z>.(2nd |` (_V X. _V))v <-> <.<.x, z>., v>. e. (2nd |` (_V X. _V)))
34		df2nd2 5069	. . . . . . . . . . . . . . 15 |- {<.<.u, t>., s>. | s = t} = (2nd |` (_V X. _V))
35	34	eleq2i 1961	. . . . . . . . . . . . . 14 |- (<.<.x, z>., v>. e. {<.<.u, t>., s>. | s = t} <-> <.<.x, z>., v>. e. (2nd |` (_V X. _V)))
36		visset 2295	. . . . . . . . . . . . . . 15 |- v e. _V
37		biidd 188	. . . . . . . . . . . . . . . 16 |- (u = x -> (s = t <-> s = t))
38		equequ2 1495	. . . . . . . . . . . . . . . 16 |- (t = z -> (s = t <-> s = z))
39		equequ1 1494	. . . . . . . . . . . . . . . 16 |- (s = v -> (s = z <-> v = z))
40	37, 38, 39	eloprabg 4936	. . . . . . . . . . . . . . 15 |- ((x e. _V /\ z e. _V /\ v e. _V) -> (<.<.x, z>., v>. e. {<.<.u, t>., s>. | s = t} <-> v = z))
41	24, 22, 36, 40	mp3an 1191	. . . . . . . . . . . . . 14 |- (<.<.x, z>., v>. e. {<.<.u, t>., s>. | s = t} <-> v = z)
42	33, 35, 41	3bitr2i 196	. . . . . . . . . . . . 13 |- (<.x, z>.(2nd |` (_V X. _V))v <-> v = z)
43		df-br 3339	. . . . . . . . . . . . . 14 |- (v({(G` y)} X. (_V X. {y}))w <-> <.v, w>. e. ({(G` y)} X. (_V X. {y})))
44	29	opelxp 4036	. . . . . . . . . . . . . 14 |- (<.v, w>. e. ({(G` y)} X. (_V X. {y})) <-> (v e. {(G` y)} /\ w e. (_V X. {y})))
45		elsn 3058	. . . . . . . . . . . . . . 15 |- (v e. {(G` y)} <-> v = (G` y))
46	45	anbi1i 539	. . . . . . . . . . . . . 14 |- ((v e. {(G` y)} /\ w e. (_V X. {y})) <-> (v = (G` y) /\ w e. (_V X. {y})))
47	43, 44, 46	3bitri 194	. . . . . . . . . . . . 13 |- (v({(G` y)} X. (_V X. {y}))w <-> (v = (G` y) /\ w e. (_V X. {y})))
48	42, 47	anbi12i 540	. . . . . . . . . . . 12 |- ((<.x, z>.(2nd |` (_V X. _V))v /\ v({(G` y)} X. (_V X. {y}))w) <-> (v = z /\ (v = (G` y) /\ w e. (_V X. {y}))))
49	48	exbii 1398	. . . . . . . . . . 11 |- (E.v(<.x, z>.(2nd |` (_V X. _V))v /\ v({(G` y)} X. (_V X. {y}))w) <-> E.v(v = z /\ (v = (G` y) /\ w e. (_V X. {y}))))
50		eqeq1 1890	. . . . . . . . . . . . 13 |- (v = z -> (v = (G` y) <-> z = (G` y)))
51	50	anbi1d 679	. . . . . . . . . . . 12 |- (v = z -> ((v = (G` y) /\ w e. (_V X. {y})) <-> (z = (G` y) /\ w e. (_V X. {y}))))
52	22, 51	ceqsexv 2325	. . . . . . . . . . 11 |- (E.v(v = z /\ (v = (G` y) /\ w e. (_V X. {y}))) <-> (z = (G` y) /\ w e. (_V X. {y})))
53	32, 49, 52	3bitri 194	. . . . . . . . . 10 |- (<.<.x, z>., w>. e. (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) <-> (z = (G` y) /\ w e. (_V X. {y})))
54	28, 30, 53	3bitr4ri 201	. . . . . . . . 9 |- (<.<.x, z>., w>. e. (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) <-> <.<.x, z>., w>. e. ((_V X. {(G` y)}) X. (_V X. {y})))
55	54	ax-gen 1305	. . . . . . . 8 |- A.w(<.<.x, z>., w>. e. (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) <-> <.<.x, z>., w>. e. ((_V X. {(G` y)}) X. (_V X. {y})))
56	55	gen2 1329	. . . . . . 7 |- A.xA.zA.w(<.<.x, z>., w>. e. (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) <-> <.<.x, z>., w>. e. ((_V X. {(G` y)}) X. (_V X. {y})))
57		relco 4392	. . . . . . . . . . 11 |- Rel (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V)))
58		dmcoss 4211	. . . . . . . . . . . . 13 |- dom (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) C_ dom (2nd |` (_V X. _V))
59		dmres 4234	. . . . . . . . . . . . . 14 |- dom (2nd |` (_V X. _V)) = ((_V X. _V) i^i dom 2nd)
60		inss1 2812	. . . . . . . . . . . . . 14 |- ((_V X. _V) i^i dom 2nd) C_ (_V X. _V)
61	59, 60	eqsstri 2647	. . . . . . . . . . . . 13 |- dom (2nd |` (_V X. _V)) C_ (_V X. _V)
62	58, 61	sstri 2626	. . . . . . . . . . . 12 |- dom (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) C_ (_V X. _V)
63		relxp 4088	. . . . . . . . . . . 12 |- Rel (_V X. _V)
64		relss 4074	. . . . . . . . . . . 12 |- (dom (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) C_ (_V X. _V) -> (Rel (_V X. _V) -> Rel dom (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V)))))
65	62, 63, 64	mp2 54	. . . . . . . . . . 11 |- Rel dom (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V)))
66	57, 65	pm3.2i 307	. . . . . . . . . 10 |- (Rel (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) /\ Rel dom (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))))
67		relrelss 4417	. . . . . . . . . 10 |- ((Rel (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) /\ Rel dom (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V)))) <-> (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) C_ ((_V X. _V) X. _V))
68	66, 67	mpbi 206	. . . . . . . . 9 |- (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) C_ ((_V X. _V) X. _V)
69		relxp 4088	. . . . . . . . . . 11 |- Rel ((_V X. {(G` y)}) X. (_V X. {y}))
70		dmxpss 4343	. . . . . . . . . . . 12 |- dom ((_V X. {(G` y)}) X. (_V X. {y})) C_ (_V X. {(G` y)})
71		relxp 4088	. . . . . . . . . . . 12 |- Rel (_V X. {(G` y)})
72		relss 4074	. . . . . . . . . . . 12 |- (dom ((_V X. {(G` y)}) X. (_V X. {y})) C_ (_V X. {(G` y)}) -> (Rel (_V X. {(G` y)}) -> Rel dom ((_V X. {(G` y)}) X. (_V X. {y}))))
73	70, 71, 72	mp2 54	. . . . . . . . . . 11 |- Rel dom ((_V X. {(G` y)}) X. (_V X. {y}))
74	69, 73	pm3.2i 307	. . . . . . . . . 10 |- (Rel ((_V X. {(G` y)}) X. (_V X. {y})) /\ Rel dom ((_V X. {(G` y)}) X. (_V X. {y})))
75		relrelss 4417	. . . . . . . . . 10 |- ((Rel ((_V X. {(G` y)}) X. (_V X. {y})) /\ Rel dom ((_V X. {(G` y)}) X. (_V X. {y}))) <-> ((_V X. {(G` y)}) X. (_V X. {y})) C_ ((_V X. _V) X. _V))
76	74, 75	mpbi 206	. . . . . . . . 9 |- ((_V X. {(G` y)}) X. (_V X. {y})) C_ ((_V X. _V) X. _V)
77	68, 76	unssi 2781	. . . . . . . 8 |- ((({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) u. ((_V X. {(G` y)}) X. (_V X. {y}))) C_ ((_V X. _V) X. _V)
78		eqrelrel 4085	. . . . . . . 8 |- (((({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) u. ((_V X. {(G` y)}) X. (_V X. {y}))) C_ ((_V X. _V) X. _V) -> ((({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) = ((_V X. {(G` y)}) X. (_V X. {y})) <-> A.xA.zA.w(<.<.x, z>., w>. e. (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) <-> <.<.x, z>., w>. e. ((_V X. {(G` y)}) X. (_V X. {y})))))
79	77, 78	ax-mp 7	. . . . . . 7 |- ((({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) = ((_V X. {(G` y)}) X. (_V X. {y})) <-> A.xA.zA.w(<.<.x, z>., w>. e. (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) <-> <.<.x, z>., w>. e. ((_V X. {(G` y)}) X. (_V X. {y}))))
80	56, 79	mpbir 207	. . . . . 6 |- (({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) = ((_V X. {(G` y)}) X. (_V X. {y}))
81	80	cnveqi 4136	. . . . 5 |- `'(({(G` y)} X. (_V X. {y})) o. (2nd |` (_V X. _V))) = `'((_V X. {(G` y)}) X. (_V X. {y}))
82		cnvxp 4332	. . . . 5 |- `'((_V X. {(G` y)}) X. (_V X. {y})) = ((_V X. {y}) X. (_V X. {(G` y)}))
83	21, 81, 82	3eqtri 1912	. . . 4 |- (`'(2nd |` (_V X. _V)) o. ((_V X. {y}) X. {(G` y)})) = ((_V X. {y}) X. (_V X. {(G` y)}))
84	83	a1i 8	. . 3 |- ((G Fn B /\ y e. B) -> (`'(2nd |` (_V X. _V)) o. ((_V X. {y}) X. {(G` y)})) = ((_V X. {y}) X. (_V X. {(G` y)})))
85	84	iuneq2dv 3279	. 2 |- (G Fn B -> U_y e. B (`'(2nd |` (_V X. _V)) o. ((_V X. {y}) X. {(G` y)})) = U_y e. B ((_V X. {y}) X. (_V X. {(G` y)})))
86	15, 17, 85	3eqtrd 1929	1 |- (G Fn B -> (`'(2nd |` (_V X. _V)) o. (G o. (2nd |` (_V X. _V)))) = U_y e. B ((_V X. {y}) X. (_V X. {(G` y)})))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292   u. cun 2591   i^i cin 2592   C_ wss 2593  {csn 3044  <.cop 3046  U_ciun 3255   class class class wbr 3338   X. cxp 3984  `'ccnv 3985  dom cdm 3986  ran crn 3987   |` cres 3988  "cima 3989   o. ccom 3990  Rel wrel 3991   Fn wfn 3993  ` cfv 3998  {copab2 4885  2ndc2nd 5019

This theorem is referenced by:  fpar 5085

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-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-f 4010  df-fo 4012  df-fv 4014  df-oprab 4887  df-1st 5020  df-2nd 5021

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