Theorem oprabco 10159

Description: Composition of a function with an operator abstraction. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
oprabco.1	|- ((x e. A /\ y e. B) -> C e. D)
oprabco.2	|- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}
oprabco.3	|- G = {<.<.x, y>., w>. | ((x e. A /\ y e. B) /\ w = (H` C))}

Assertion

Ref	Expression
oprabco	|- (H Fn D -> G = (H o. F))

Distinct variable groups:   w,A,x,y,z   w,B,x,y,z   w,C,z   w,D,x,y,z   z,G   w,H,x,y,z

Proof of Theorem oprabco

Step	Hyp	Ref	Expression
1		eqfnfv 4766	. . 3 |- ((G Fn (A X. B) /\ (H o. F) Fn (A X. B)) -> (G = (H o. F) <-> ((A X. B) = (A X. B) /\ A.u e. (A X. B)(G` u) = ((H o. F)` u))))
2		fvex 4689	. . . 4 |- (H` C) e. _V
3		oprabco.3	. . . 4 |- G = {<.<.x, y>., w>. | ((x e. A /\ y e. B) /\ w = (H` C))}
4	2, 3	fnoprab2 5064	. . 3 |- G Fn (A X. B)
5		oprabco.2	. . . . . 6 |- F = {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)}
6		oprabco.1	. . . . . 6 |- ((x e. A /\ y e. B) -> C e. D)
7	5, 6	foprab 5062	. . . . 5 |- F:(A X. B)-->D
8		ffn 4562	. . . . 5 |- (F:(A X. B)-->D -> F Fn (A X. B))
9	7, 8	ax-mp 7	. . . 4 |- F Fn (A X. B)
10		frn 4569	. . . . 5 |- (F:(A X. B)-->D -> ran F C_ D)
11	7, 10	ax-mp 7	. . . 4 |- ran F C_ D
12		fnco 4521	. . . 4 |- ((H Fn D /\ F Fn (A X. B) /\ ran F C_ D) -> (H o. F) Fn (A X. B))
13	9, 11, 12	mp3an23 1183	. . 3 |- (H Fn D -> (H o. F) Fn (A X. B))
14	1, 4, 13	sylancr 526	. 2 |- (H Fn D -> (G = (H o. F) <-> ((A X. B) = (A X. B) /\ A.u e. (A X. B)(G` u) = ((H o. F)` u))))
15		eqidd 1885	. 2 |- (H Fn D -> (A X. B) = (A X. B))
16		xp1st 10155	. . . . . . . . 9 |- (u e. (A X. B) -> (1st` u) e. A)
17		xp2nd 10156	. . . . . . . . 9 |- (u e. (A X. B) -> (2nd` u) e. B)
18	6	rgen2 2186	. . . . . . . . . 10 |- A.x e. A A.y e. B C e. D
19		ax-17 1317	. . . . . . . . . . . . 13 |- (y e. B -> A.x y e. B)
20		fvex 4689	. . . . . . . . . . . . . . 15 |- (1st` u) e. _V
21		ax-17 1317	. . . . . . . . . . . . . . 15 |- (v e. (1st` u) -> A.x v e. (1st` u))
22	20, 21	hbcsb1 2568	. . . . . . . . . . . . . 14 |- (v e. [_(1st` u) / x]_C -> A.x v e. [_(1st` u) / x]_C)
23		ax-17 1317	. . . . . . . . . . . . . 14 |- (v e. D -> A.x v e. D)
24	22, 23	hbel 1996	. . . . . . . . . . . . 13 |- ([_(1st` u) / x]_C e. D -> A.x[_(1st` u) / x]_C e. D)
25	19, 24	hbral 2146	. . . . . . . . . . . 12 |- (A.y e. B [_(1st` u) / x]_C e. D -> A.xA.y e. B [_(1st` u) / x]_C e. D)
26		csbeq1a 2546	. . . . . . . . . . . . . 14 |- (x = (1st` u) -> C = [_(1st` u) / x]_C)
27	26	eleq1d 1963	. . . . . . . . . . . . 13 |- (x = (1st` u) -> (C e. D <-> [_(1st` u) / x]_C e. D))
28	27	ralbidv 2123	. . . . . . . . . . . 12 |- (x = (1st` u) -> (A.y e. B C e. D <-> A.y e. B [_(1st` u) / x]_C e. D))
29	25, 28	rcla4 2373	. . . . . . . . . . 11 |- ((1st` u) e. A -> (A.x e. A A.y e. B C e. D -> A.y e. B [_(1st` u) / x]_C e. D))
30		fvex 4689	. . . . . . . . . . . . . 14 |- (2nd` u) e. _V
31		ax-17 1317	. . . . . . . . . . . . . 14 |- (v e. (2nd` u) -> A.y v e. (2nd` u))
32	30, 31	hbcsb1 2568	. . . . . . . . . . . . 13 |- (v e. [_(2nd` u) / y]_[_(1st` u) / x]_C -> A.y v e. [_(2nd` u) / y]_[_(1st` u) / x]_C)
33		ax-17 1317	. . . . . . . . . . . . 13 |- (v e. D -> A.y v e. D)
34	32, 33	hbel 1996	. . . . . . . . . . . 12 |- ([_(2nd` u) / y]_[_(1st` u) / x]_C e. D -> A.y[_(2nd` u) / y]_[_(1st` u) / x]_C e. D)
35		csbeq1a 2546	. . . . . . . . . . . . 13 |- (y = (2nd` u) -> [_(1st` u) / x]_C = [_(2nd` u) / y]_[_(1st` u) / x]_C)
36	35	eleq1d 1963	. . . . . . . . . . . 12 |- (y = (2nd` u) -> ([_(1st` u) / x]_C e. D <-> [_(2nd` u) / y]_[_(1st` u) / x]_C e. D))
37	34, 36	rcla4 2373	. . . . . . . . . . 11 |- ((2nd` u) e. B -> (A.y e. B [_(1st` u) / x]_C e. D -> [_(2nd` u) / y]_[_(1st` u) / x]_C e. D))
38	29, 37	sylan9 517	. . . . . . . . . 10 |- (((1st` u) e. A /\ (2nd` u) e. B) -> (A.x e. A A.y e. B C e. D -> [_(2nd` u) / y]_[_(1st` u) / x]_C e. D))
39	18, 38	mpi 55	. . . . . . . . 9 |- (((1st` u) e. A /\ (2nd` u) e. B) -> [_(2nd` u) / y]_[_(1st` u) / x]_C e. D)
40	16, 17, 39	syl11anc 524	. . . . . . . 8 |- (u e. (A X. B) -> [_(2nd` u) / y]_[_(1st` u) / x]_C e. D)
41		fveq2 4681	. . . . . . . . . . 11 |- (v = u -> (2nd` v) = (2nd` u))
42	41	csbeq1d 2544	. . . . . . . . . 10 |- (v = u -> [_(2nd` v) / y]_[_(1st` v) / x]_C = [_(2nd` u) / y]_[_(1st` v) / x]_C)
43		fveq2 4681	. . . . . . . . . . . . 13 |- (v = u -> (1st` v) = (1st` u))
44	43	csbeq1d 2544	. . . . . . . . . . . 12 |- (v = u -> [_(1st` v) / x]_C = [_(1st` u) / x]_C)
45	44	csbeq2dv 2562	. . . . . . . . . . 11 |- ((v = u /\ (2nd` u) e. _V) -> [_(2nd` u) / y]_[_(1st` v) / x]_C = [_(2nd` u) / y]_[_(1st` u) / x]_C)
46	30, 45	mpan2 760	. . . . . . . . . 10 |- (v = u -> [_(2nd` u) / y]_[_(1st` v) / x]_C = [_(2nd` u) / y]_[_(1st` u) / x]_C)
47	42, 46	eqtrd 1925	. . . . . . . . 9 |- (v = u -> [_(2nd` v) / y]_[_(1st` v) / x]_C = [_(2nd` u) / y]_[_(1st` u) / x]_C)
48		ax-17 1317	. . . . . . . . . . . 12 |- (u e. z -> A.x u e. z)
49		fvex 4689	. . . . . . . . . . . . 13 |- (1st` v) e. _V
50		ax-17 1317	. . . . . . . . . . . . 13 |- (u e. (1st` v) -> A.x u e. (1st` v))
51	49, 50	hbcsb1 2568	. . . . . . . . . . . 12 |- (u e. [_(1st` v) / x]_C -> A.x u e. [_(1st` v) / x]_C)
52	48, 51	hbeq 1995	. . . . . . . . . . 11 |- (z = [_(1st` v) / x]_C -> A.x z = [_(1st` v) / x]_C)
53		ax-17 1317	. . . . . . . . . . . 12 |- (u e. z -> A.y u e. z)
54		fvex 4689	. . . . . . . . . . . . 13 |- (2nd` v) e. _V
55		ax-17 1317	. . . . . . . . . . . . 13 |- (u e. (2nd` v) -> A.y u e. (2nd` v))
56	54, 55	hbcsb1 2568	. . . . . . . . . . . 12 |- (u e. [_(2nd` v) / y]_[_(1st` v) / x]_C -> A.y u e. [_(2nd` v) / y]_[_(1st` v) / x]_C)
57	53, 56	hbeq 1995	. . . . . . . . . . 11 |- (z = [_(2nd` v) / y]_[_(1st` v) / x]_C -> A.y z = [_(2nd` v) / y]_[_(1st` v) / x]_C)
58		csbeq1a 2546	. . . . . . . . . . . 12 |- (x = (1st` v) -> C = [_(1st` v) / x]_C)
59	58	eqeq2d 1895	. . . . . . . . . . 11 |- (x = (1st` v) -> (z = C <-> z = [_(1st` v) / x]_C))
60		csbeq1a 2546	. . . . . . . . . . . 12 |- (y = (2nd` v) -> [_(1st` v) / x]_C = [_(2nd` v) / y]_[_(1st` v) / x]_C)
61	60	eqeq2d 1895	. . . . . . . . . . 11 |- (y = (2nd` v) -> (z = [_(1st` v) / x]_C <-> z = [_(2nd` v) / y]_[_(1st` v) / x]_C))
62	52, 57, 59, 61	oprabopabf 10157	. . . . . . . . . 10 |- {<.<.x, y>., z>. | ((x e. A /\ y e. B) /\ z = C)} = {<.v, z>. | (v e. (A X. B) /\ z = [_(2nd` v) / y]_[_(1st` v) / x]_C)}
63	5, 62	eqtri 1908	. . . . . . . . 9 |- F = {<.v, z>. | (v e. (A X. B) /\ z = [_(2nd` v) / y]_[_(1st` v) / x]_C)}
64	47, 63	fvopab4g 4742	. . . . . . . 8 |- ((u e. (A X. B) /\ [_(2nd` u) / y]_[_(1st` u) / x]_C e. D) -> (F` u) = [_(2nd` u) / y]_[_(1st` u) / x]_C)
65	40, 64	mpdan 768	. . . . . . 7 |- (u e. (A X. B) -> (F` u) = [_(2nd` u) / y]_[_(1st` u) / x]_C)
66	65	adantl 424	. . . . . 6 |- ((H Fn D /\ u e. (A X. B)) -> (F` u) = [_(2nd` u) / y]_[_(1st` u) / x]_C)
67	66	fveq2d 4685	. . . . 5 |- ((H Fn D /\ u e. (A X. B)) -> (H` (F` u)) = (H` [_(2nd` u) / y]_[_(1st` u) / x]_C))
68		csbfv2g 4700	. . . . . . . . 9 |- ((1st` u) e. _V -> [_(1st` u) / x]_(H` C) = (H` [_(1st` u) / x]_C))
69	20, 68	ax-mp 7	. . . . . . . 8 |- [_(1st` u) / x]_(H` C) = (H` [_(1st` u) / x]_C)
70	69	csbeq2i 2563	. . . . . . 7 |- ((2nd` u) e. _V -> [_(2nd` u) / y]_[_(1st` u) / x]_(H` C) = [_(2nd` u) / y]_(H` [_(1st` u) / x]_C))
71	30, 70	ax-mp 7	. . . . . 6 |- [_(2nd` u) / y]_[_(1st` u) / x]_(H` C) = [_(2nd` u) / y]_(H` [_(1st` u) / x]_C)
72		csbfv2g 4700	. . . . . . 7 |- ((2nd` u) e. _V -> [_(2nd` u) / y]_(H` [_(1st` u) / x]_C) = (H` [_(2nd` u) / y]_[_(1st` u) / x]_C))
73	30, 72	ax-mp 7	. . . . . 6 |- [_(2nd` u) / y]_(H` [_(1st` u) / x]_C) = (H` [_(2nd` u) / y]_[_(1st` u) / x]_C)
74	71, 73	eqtri 1908	. . . . 5 |- [_(2nd` u) / y]_[_(1st` u) / x]_(H` C) = (H` [_(2nd` u) / y]_[_(1st` u) / x]_C)
75	67, 74	syl6eqr 1946	. . . 4 |- ((H Fn D /\ u e. (A X. B)) -> (H` (F` u)) = [_(2nd` u) / y]_[_(1st` u) / x]_(H` C))
76		fnfun 4510	. . . . . 6 |- (H Fn D -> Fun H)
77	76	adantr 425	. . . . 5 |- ((H Fn D /\ u e. (A X. B)) -> Fun H)
78		fnfun 4510	. . . . . . 7 |- (F Fn (A X. B) -> Fun F)
79	9, 78	ax-mp 7	. . . . . 6 |- Fun F
80	79	a1i 8	. . . . 5 |- ((H Fn D /\ u e. (A X. B)) -> Fun F)
81	7	fdmi 4568	. . . . . . . 8 |- dom F = (A X. B)
82	81	eleq2i 1961	. . . . . . 7 |- (u e. dom F <-> u e. (A X. B))
83	82	biimpri 169	. . . . . 6 |- (u e. (A X. B) -> u e. dom F)
84	83	adantl 424	. . . . 5 |- ((H Fn D /\ u e. (A X. B)) -> u e. dom F)
85		fvco 4736	. . . . 5 |- ((Fun H /\ Fun F /\ u e. dom F) -> ((H o. F)` u) = (H` (F` u)))
86	77, 80, 84, 85	syl111anc 1100	. . . 4 |- ((H Fn D /\ u e. (A X. B)) -> ((H o. F)` u) = (H` (F` u)))
87	20, 2	csbex 2549	. . . . . . 7 |- [_(1st` u) / x]_(H` C) e. _V
88	30, 87	csbex 2549	. . . . . 6 |- [_(2nd` u) / y]_[_(1st` u) / x]_(H` C) e. _V
89	41	csbeq1d 2544	. . . . . . . 8 |- (v = u -> [_(2nd` v) / y]_[_(1st` v) / x]_(H` C) = [_(2nd` u) / y]_[_(1st` v) / x]_(H` C))
90	43	csbeq1d 2544	. . . . . . . . . 10 |- (v = u -> [_(1st` v) / x]_(H` C) = [_(1st` u) / x]_(H` C))
91	90	csbeq2dv 2562	. . . . . . . . 9 |- ((v = u /\ (2nd` u) e. _V) -> [_(2nd` u) / y]_[_(1st` v) / x]_(H` C) = [_(2nd` u) / y]_[_(1st` u) / x]_(H` C))
92	30, 91	mpan2 760	. . . . . . . 8 |- (v = u -> [_(2nd` u) / y]_[_(1st` v) / x]_(H` C) = [_(2nd` u) / y]_[_(1st` u) / x]_(H` C))
93	89, 92	eqtrd 1925	. . . . . . 7 |- (v = u -> [_(2nd` v) / y]_[_(1st` v) / x]_(H` C) = [_(2nd` u) / y]_[_(1st` u) / x]_(H` C))
94		ax-17 1317	. . . . . . . . . 10 |- (u e. w -> A.x u e. w)
95	49, 50	hbcsb1 2568	. . . . . . . . . 10 |- (u e. [_(1st` v) / x]_(H` C) -> A.x u e. [_(1st` v) / x]_(H` C))
96	94, 95	hbeq 1995	. . . . . . . . 9 |- (w = [_(1st` v) / x]_(H` C) -> A.x w = [_(1st` v) / x]_(H` C))
97		ax-17 1317	. . . . . . . . . 10 |- (u e. w -> A.y u e. w)
98	54, 55	hbcsb1 2568	. . . . . . . . . 10 |- (u e. [_(2nd` v) / y]_[_(1st` v) / x]_(H` C) -> A.y u e. [_(2nd` v) / y]_[_(1st` v) / x]_(H` C))
99	97, 98	hbeq 1995	. . . . . . . . 9 |- (w = [_(2nd` v) / y]_[_(1st` v) / x]_(H` C) -> A.y w = [_(2nd` v) / y]_[_(1st` v) / x]_(H` C))
100		csbeq1a 2546	. . . . . . . . . 10 |- (x = (1st` v) -> (H` C) = [_(1st` v) / x]_(H` C))
101	100	eqeq2d 1895	. . . . . . . . 9 |- (x = (1st` v) -> (w = (H` C) <-> w = [_(1st` v) / x]_(H` C)))
102		csbeq1a 2546	. . . . . . . . . 10 |- (y = (2nd` v) -> [_(1st` v) / x]_(H` C) = [_(2nd` v) / y]_[_(1st` v) / x]_(H` C))
103	102	eqeq2d 1895	. . . . . . . . 9 |- (y = (2nd` v) -> (w = [_(1st` v) / x]_(H` C) <-> w = [_(2nd` v) / y]_[_(1st` v) / x]_(H` C)))
104	96, 99, 101, 103	oprabopabf 10157	. . . . . . . 8 |- {<.<.x, y>., w>. | ((x e. A /\ y e. B) /\ w = (H` C))} = {<.v, w>. | (v e. (A X. B) /\ w = [_(2nd` v) / y]_[_(1st` v) / x]_(H` C))}
105	3, 104	eqtri 1908	. . . . . . 7 |- G = {<.v, w>. | (v e. (A X. B) /\ w = [_(2nd` v) / y]_[_(1st` v) / x]_(H` C))}
106	93, 105	fvopab4g 4742	. . . . . 6 |- ((u e. (A X. B) /\ [_(2nd` u) / y]_[_(1st` u) / x]_(H` C) e. _V) -> (G` u) = [_(2nd` u) / y]_[_(1st` u) / x]_(H` C))
107	88, 106	mpan2 760	. . . . 5 |- (u e. (A X. B) -> (G` u) = [_(2nd` u) / y]_[_(1st` u) / x]_(H` C))
108	107	adantl 424	. . . 4 |- ((H Fn D /\ u e. (A X. B)) -> (G` u) = [_(2nd` u) / y]_[_(1st` u) / x]_(H` C))
109	75, 86, 108	3eqtr4rd 1939	. . 3 |- ((H Fn D /\ u e. (A X. B)) -> (G` u) = ((H o. F)` u))
110	109	r19.21aiva 2176	. 2 |- (H Fn D -> A.u e. (A X. B)(G` u) = ((H o. F)` u))
111	14, 15, 110	mpbir2and 802	1 |- (H Fn D -> G = (H o. F))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  [_csb 2540   C_ wss 2593  {copab 3395   X. cxp 3984  dom cdm 3986  ran crn 3987   o. ccom 3990  Fun wfun 3992   Fn wfn 3993  -->wf 3994  ` cfv 3998  {copab2 4885  1stc1st 5018  2ndc2nd 5019

This theorem is referenced by:  oprab2co 10160  cnoprab1 15921  cnoprab2 15922  phtpyid 16049  phtpycom 16050  phtpycolem3 16053  phtpycolem4 16054  reparpht 16065  pcorevlem 16086

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-5 1302  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-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-fv 4014  df-oprab 4887  df-1st 5020  df-2nd 5021

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