Theorem fopabcos 4806

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 C_ 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 2295	. . . . . . . 8 |- z e. _V
2		fopabcos.2	. . . . . . . 8 |- D e. _V
3		fopabcos.4	. . . . . . . 8 |- G = {<.x, y>. | (x e. B /\ y = D)}
4	1, 2, 3	fvopab4s 4746	. . . . . . 7 |- (z e. B -> (G` z) = [_z / x]_D)
5	4	adantl 424	. . . . . 6 |- ((ran G C_ A /\ z e. B) -> (G` z) = [_z / x]_D)
6	2, 3	fnopab2 4549	. . . . . . . . 9 |- G Fn B
7		fnfvelrn 4786	. . . . . . . . 9 |- ((G Fn B /\ z e. B) -> (G` z) e. ran G)
8	6, 7	mpan 759	. . . . . . . 8 |- (z e. B -> (G` z) e. ran G)
9	8	adantl 424	. . . . . . 7 |- ((ran G C_ A /\ z e. B) -> (G` z) e. ran G)
10		ssel 2615	. . . . . . . 8 |- (ran G C_ A -> ((G` z) e. ran G -> (G` z) e. A))
11	10	adantr 425	. . . . . . 7 |- ((ran G C_ A /\ z e. B) -> ((G` z) e. ran G -> (G` z) e. A))
12	9, 11	mpd 29	. . . . . 6 |- ((ran G C_ A /\ z e. B) -> (G` z) e. A)
13	5, 12	eqeltrrd 1972	. . . . 5 |- ((ran G C_ A /\ z e. B) -> [_z / x]_D e. A)
14	1, 2	csbex 2549	. . . . . 6 |- [_z / x]_D e. _V
15		fopabcos.1	. . . . . 6 |- C e. _V
16		ax-17 1317	. . . . . . 7 |- (w e. z -> A.x w e. z)
17	1, 16	hbcsb1 2568	. . . . . 6 |- (w e. [_z / x]_D -> A.x w e. [_z / x]_D)
18		fopabcos.3	. . . . . 6 |- F = {<.x, y>. | (x e. A /\ y = C)}
19	14, 15, 17, 18	fvopab4sf 4745	. . . . 5 |- ([_z / x]_D e. A -> (F` [_z / x]_D) = [_[_z / x]_D / x]_C)
20	13, 19	syl 12	. . . 4 |- ((ran G C_ A /\ z e. B) -> (F` [_z / x]_D) = [_[_z / x]_D / x]_C)
21	2, 3	dmopab2 4550	. . . . . . . 8 |- dom G = B
22	21	eleq2i 1961	. . . . . . 7 |- (z e. dom G <-> z e. B)
23	15, 18	fnopab2 4549	. . . . . . . . 9 |- F Fn A
24		fnfun 4510	. . . . . . . . 9 |- (F Fn A -> Fun F)
25	23, 24	ax-mp 7	. . . . . . . 8 |- Fun F
26		fnfun 4510	. . . . . . . . 9 |- (G Fn B -> Fun G)
27	6, 26	ax-mp 7	. . . . . . . 8 |- Fun G
28		fvco 4736	. . . . . . . 8 |- ((Fun F /\ Fun G /\ z e. dom G) -> ((F o. G)` z) = (F` (G` z)))
29	25, 27, 28	mp3an12 1181	. . . . . . 7 |- (z e. dom G -> ((F o. G)` z) = (F` (G` z)))
30	22, 29	sylbir 218	. . . . . 6 |- (z e. B -> ((F o. G)` z) = (F` (G` z)))
31	4	fveq2d 4685	. . . . . 6 |- (z e. B -> (F` (G` z)) = (F` [_z / x]_D))
32	30, 31	eqtrd 1925	. . . . 5 |- (z e. B -> ((F o. G)` z) = (F` [_z / x]_D))
33	32	adantl 424	. . . 4 |- ((ran G C_ A /\ z e. B) -> ((F o. G)` z) = (F` [_z / x]_D))
34	2, 15	csbex 2549	. . . . . . 7 |- [_D / x]_C e. _V
35		eqid 1884	. . . . . . 7 |- {<.x, y>. | (x e. B /\ y = [_D / x]_C)} = {<.x, y>. | (x e. B /\ y = [_D / x]_C)}
36	1, 34, 35	fvopab4s 4746	. . . . . 6 |- (z e. B -> ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z) = [_z / x]_[_D / x]_C)
37	2	ax-gen 1305	. . . . . . 7 |- A.x D e. _V
38		csbnest1g 2582	. . . . . . 7 |- ((z e. _V /\ A.x D e. _V) -> [_z / x]_[_D / x]_C = [_[_z / x]_D / x]_C)
39	1, 37, 38	mp2an 761	. . . . . 6 |- [_z / x]_[_D / x]_C = [_[_z / x]_D / x]_C
40	36, 39	syl6eq 1944	. . . . 5 |- (z e. B -> ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z) = [_[_z / x]_D / x]_C)
41	40	adantl 424	. . . 4 |- ((ran G C_ A /\ z e. B) -> ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z) = [_[_z / x]_D / x]_C)
42	20, 33, 41	3eqtr4d 1937	. . 3 |- ((ran G C_ A /\ z e. B) -> ((F o. G)` z) = ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z))
43	42	r19.21aiva 2176	. 2 |- (ran G C_ A -> A.z e. B ((F o. G)` z) = ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z))
44		ax-17 1317	. . . 4 |- (w e. (F o. G) -> A.z w e. (F o. G))
45		ax-17 1317	. . . 4 |- (w e. {<.x, y>. | (x e. B /\ y = [_D / x]_C)} -> A.z w e. {<.x, y>. | (x e. B /\ y = [_D / x]_C)})
46	44, 45	eqfnfv2f 4770	. . 3 |- (((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)} <-> A.z e. B ((F o. G)` z) = ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z)))
47		fnco 4521	. . . 4 |- ((F Fn A /\ G Fn B /\ ran G C_ A) -> (F o. G) Fn B)
48	23, 6, 47	mp3an12 1181	. . 3 |- (ran G C_ A -> (F o. G) Fn B)
49	34, 35	fnopab2 4549	. . 3 |- {<.x, y>. | (x e. B /\ y = [_D / x]_C)} Fn B
50	46, 48, 49	sylancl 525	. 2 |- (ran G C_ A -> ((F o. G) = {<.x, y>. | (x e. B /\ y = [_D / x]_C)} <-> A.z e. B ((F o. G)` z) = ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z)))
51	43, 50	mpbird 213	1 |- (ran G C_ A -> (F o. G) = {<.x, y>. | (x e. B /\ y = [_D / x]_C)})

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292  [_csb 2540   C_ wss 2593  {copab 3395  dom cdm 3986  ran crn 3987   o. ccom 3990  Fun wfun 3992   Fn wfn 3993  ` cfv 3998

This theorem is referenced by:  oprcn 9255  kbass2 11688  kbass5 11691

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-fv 4014

Copyright terms: Public domain