Theorem fopabcos 3909

Description: Composition of two functions expressed as ordered-pair class abstractions.

Hypotheses

Ref	Expression
fopabcos.1	|- C e. V
fopabcos.2	|- D e. V
fopabcos.3	|- F = {<.x, y>. | (x e. A /\ y = C)}
fopabcos.4	|- G = {<.x, y>. | (x e. B /\ y = D)}

Assertion

Ref	Expression
fopabcos	|- (ran G (_ A -> (F o. G) = {<.x, y>. | (x e. B /\ y = [_D / x]_C)})

Distinct variable groups:   x,y,A   x,B,y   y,C   y,D

Proof of Theorem fopabcos

Step	Hyp	Ref	Expression
1		visset 1851	. . . . . . . . 9 |- z e. V
2		fopabcos.2	. . . . . . . . 9 |- D e. V
3		fopabcos.4	. . . . . . . . 9 |- G = {<.x, y>. | (x e. B /\ y = D)}
4	1, 2, 3	fvopab4s 3859	. . . . . . . 8 |- (z e. B -> (G` z) = [_z / x]_D)
5	4	adantl 388	. . . . . . 7 |- ((ran G (_ A /\ z e. B) -> (G` z) = [_z / x]_D)
6	2, 3	fnopab2 3693	. . . . . . . . . 10 |- G Fn B
7		fnfvelrn 3889	. . . . . . . . . 10 |- ((G Fn B /\ z e. B) -> (G` z) e. ran G)
8	6, 7	mpan 698	. . . . . . . . 9 |- (z e. B -> (G` z) e. ran G)
9	8	adantl 388	. . . . . . . 8 |- ((ran G (_ A /\ z e. B) -> (G` z) e. ran G)
10		ssel 2107	. . . . . . . . 9 |- (ran G (_ A -> ((G` z) e. ran G -> (G` z) e. A))
11	10	adantr 389	. . . . . . . 8 |- ((ran G (_ A /\ z e. B) -> ((G` z) e. ran G -> (G` z) e. A))
12	9, 11	mpd 26	. . . . . . 7 |- ((ran G (_ A /\ z e. B) -> (G` z) e. A)
13	5, 12	eqeltrrd 1586	. . . . . 6 |- ((ran G (_ A /\ z e. B) -> [_z / x]_D e. A)
14	1, 2	csbex 2052	. . . . . . 7 |- [_z / x]_D e. V
15		fopabcos.1	. . . . . . 7 |- C e. V
16		ax-17 1003	. . . . . . . 8 |- (w e. z -> A.x w e. z)
17	1, 16	hbcsb1 2068	. . . . . . 7 |- (w e. [_z / x]_D -> A.x w e. [_z / x]_D)
18		fopabcos.3	. . . . . . 7 |- F = {<.x, y>. | (x e. A /\ y = C)}
19	14, 15, 17, 18	fvopab4sf 3858	. . . . . 6 |- ([_z / x]_D e. A -> (F` [_z / x]_D) = [_[_z / x]_D / x]_C)
20	13, 19	syl 10	. . . . 5 |- ((ran G (_ A /\ z e. B) -> (F` [_z / x]_D) = [_[_z / x]_D / x]_C)
21	2, 3	dmopab2 3694	. . . . . . . . 9 |- dom G = B
22	21	eleq2i 1575	. . . . . . . 8 |- (z e. dom G <-> z e. B)
23	15, 18	fnopab2 3693	. . . . . . . . . 10 |- F Fn A
24		fnfun 3660	. . . . . . . . . 10 |- (F Fn A -> Fun F)
25	23, 24	ax-mp 7	. . . . . . . . 9 |- Fun F
26		fnfun 3660	. . . . . . . . . 10 |- (G Fn B -> Fun G)
27	6, 26	ax-mp 7	. . . . . . . . 9 |- Fun G
28		fvco 3850	. . . . . . . . 9 |- ((Fun F /\ Fun G /\ z e. dom G) -> ((F o. G)` z) = (F` (G` z)))
29	25, 27, 28	mp3an12 909	. . . . . . . 8 |- (z e. dom G -> ((F o. G)` z) = (F` (G` z)))
30	22, 29	sylbir 199	. . . . . . 7 |- (z e. B -> ((F o. G)` z) = (F` (G` z)))
31	4	fveq2d 3804	. . . . . . 7 |- (z e. B -> (F` (G` z)) = (F` [_z / x]_D))
32	30, 31	eqtrd 1544	. . . . . 6 |- (z e. B -> ((F o. G)` z) = (F` [_z / x]_D))
33	32	adantl 388	. . . . 5 |- ((ran G (_ A /\ z e. B) -> ((F o. G)` z) = (F` [_z / x]_D))
34	2, 15	csbex 2052	. . . . . . . 8 |- [_D / x]_C e. V
35		eqid 1512	. . . . . . . 8 |- {<.x, y>. | (x e. B /\ y = [_D / x]_C)} = {<.x, y>. | (x e. B /\ y = [_D / x]_C)}
36	1, 34, 35	fvopab4s 3859	. . . . . . 7 |- (z e. B -> ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z) = [_z / x]_[_D / x]_C)
37	2	ax-gen 995	. . . . . . . 8 |- A.x D e. V
38		csbnest1g 2081	. . . . . . . 8 |- ((z e. V /\ A.x D e. V) -> [_z / x]_[_D / x]_C = [_[_z / x]_D / x]_C)
39	1, 37, 38	mp2an 700	. . . . . . 7 |- [_z / x]_[_D / x]_C = [_[_z / x]_D / x]_C
40	36, 39	syl6eq 1560	. . . . . 6 |- (z e. B -> ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z) = [_[_z / x]_D / x]_C)
41	40	adantl 388	. . . . 5 |- ((ran G (_ A /\ z e. B) -> ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z) = [_[_z / x]_D / x]_C)
42	20, 33, 41	3eqtr4d 1554	. . . 4 |- ((ran G (_ A /\ z e. B) -> ((F o. G)` z) = ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z))
43	42	r19.21aiva 1752	. . 3 |- (ran G (_ A -> A.z e. B ((F o. G)` z) = ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z))
44		eqid 1512	. . 3 |- B = B
45	43, 44	jctil 290	. 2 |- (ran G (_ A -> (B = B /\ A.z e. B ((F o. G)` z) = ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z)))
46		fnco 3670	. . . 4 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> (F o. G) Fn B)
47	23, 6, 46	mp3an12 909	. . 3 |- (ran G (_ A -> (F o. G) Fn B)
48	34, 35	fnopab2 3693	. . . 4 |- {<.x, y>. | (x e. B /\ y = [_D / x]_C)} Fn B
49		ax-17 1003	. . . . 5 |- (w e. (F o. G) -> A.z w e. (F o. G))
50		ax-17 1003	. . . . 5 |- (w e. {<.x, y>. | (x e. B /\ y = [_D / x]_C)} -> A.z w e. {<.x, y>. | (x e. B /\ y = [_D / x]_C)})
51	49, 50	eqfnfvf 3874	. . . 4 |- (((F o. G) Fn B /\ {<.x, y>. | (x e. B /\ y = [_D / x]_C)} Fn B) -> ((F o. G) = {<.x, y>. | (x e. B /\ y = [_D / x]_C)} <-> (B = B /\ A.z e. B ((F o. G)` z) = ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z))))
52	48, 51	mpan2 699	. . 3 |- ((F o. G) Fn B -> ((F o. G) = {<.x, y>. | (x e. B /\ y = [_D / x]_C)} <-> (B = B /\ A.z e. B ((F o. G)` z) = ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z))))
53	47, 52	syl 10	. 2 |- (ran G (_ A -> ((F o. G) = {<.x, y>. | (x e. B /\ y = [_D / x]_C)} <-> (B = B /\ A.z e. B ((F o. G)` z) = ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z))))
54	45, 53	mpbird 194	1 |- (ran G (_ A -> (F o. G) = {<.x, y>. | (x e. B /\ y = [_D / x]_C)})

Syntax hints:   -> wi 3   <-> wb 144   /\ wa 221  A.wal 986   = wceq 988   e. wcel 990  A.wral 1683  Vcvv 1849  [_csb 2043   (_ wss 2091  {copab 2717  dom cdm 3225  ran crn 3226   o. ccom 3229  Fun wfun 3231   Fn wfn 3232  ` cfv 3237

This theorem is referenced by:  oprcn 8097  kbass2 10167  kbass5 10170

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-sep 2754  ax-pow 2794  ax-pr 2832  ax-un 2920

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-ral 1687  df-rex 1688  df-v 1850  df-sbc 1979  df-csb 2044  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-fv 3253

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