Theorem fopabcos 3772

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 1788	. . . . . . . . 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 3722	. . . . . . . 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 3558	. . . . . . . . . 10 |- G Fn B
7		fnfvelrn 3752	. . . . . . . . . 10 |- ((G Fn B /\ z e. B) -> (G` z) e. ran G)
8	6, 7	mpan 692	. . . . . . . . 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 2034	. . . . . . . . 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 1525	. . . . . 6 |- ((ran G (_ A /\ z e. B) -> [_z / x]_D e. A)
14	1, 2	csbex 1980	. . . . . . 7 |- [_z / x]_D e. V
15		fopabcos.1	. . . . . . 7 |- C e. V
16		ax-17 1190	. . . . . . . 8 |- (w e. z -> A.x w e. z)
17	1, 16	hbcsb1 1996	. . . . . . 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 3721	. . . . . 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 3559	. . . . . . . . 9 |- dom G = B
22	21	eleq2i 1514	. . . . . . . 8 |- (z e. dom G <-> z e. B)
23	15, 18	fnopab2 3558	. . . . . . . . . 10 |- F Fn A
24		fnfun 3525	. . . . . . . . . 10 |- (F Fn A -> Fun F)
25	23, 24	ax-mp 7	. . . . . . . . 9 |- Fun F
26		fnfun 3525	. . . . . . . . . 10 |- (G Fn B -> Fun G)
27	6, 26	ax-mp 7	. . . . . . . . 9 |- Fun G
28		fvco 3713	. . . . . . . . 9 |- ((Fun F /\ Fun G /\ z e. dom G) -> ((F o. G)` z) = (F` (G` z)))
29	25, 27, 28	mp3an12 902	. . . . . . . 8 |- (z e. dom G -> ((F o. G)` z) = (F` (G` z)))
30	22, 29	sylbir 201	. . . . . . 7 |- (z e. B -> ((F o. G)` z) = (F` (G` z)))
31	4	fveq2d 3667	. . . . . . 7 |- (z e. B -> (F` (G` z)) = (F` [_z / x]_D))
32	30, 31	eqtrd 1483	. . . . . 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 1980	. . . . . . . 8 |- [_D / x]_C e. V
35		eqid 1452	. . . . . . . 8 |- {<.x, y>. | (x e. B /\ y = [_D / x]_C)} = {<.x, y>. | (x e. B /\ y = [_D / x]_C)}
36	1, 34, 35	fvopab4s 3722	. . . . . . 7 |- (z e. B -> ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z) = [_z / x]_[_D / x]_C)
37	2	ax-gen 955	. . . . . . . 8 |- A.x D e. V
38		csbnest1g 2008	. . . . . . . 8 |- ((z e. V /\ A.x D e. V) -> [_z / x]_[_D / x]_C = [_[_z / x]_D / x]_C)
39	1, 37, 38	mp2an 694	. . . . . . 7 |- [_z / x]_[_D / x]_C = [_[_z / x]_D / x]_C
40	36, 39	syl6eq 1499	. . . . . 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 1493	. . . 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 1690	. . 3 |- (ran G (_ A -> A.z e. B ((F o. G)` z) = ({<.x, y>. | (x e. B /\ y = [_D / x]_C)}` z))
44		eqid 1452	. . 3 |- B = B
45	43, 44	jctil 292	. 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 3535	. . . 4 |- ((F Fn A /\ G Fn B /\ ran G (_ A) -> (F o. G) Fn B)
47	23, 6, 46	mp3an12 902	. . 3 |- (ran G (_ A -> (F o. G) Fn B)
48	34, 35	fnopab2 3558	. . . 4 |- {<.x, y>. | (x e. B /\ y = [_D / x]_C)} Fn B
49		ax-17 1190	. . . . 5 |- (w e. (F o. G) -> A.z w e. (F o. G))
50		ax-17 1190	. . . . 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 3737	. . . 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 693	. . 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 196	1 |- (ran G (_ A -> (F o. G) = {<.x, y>. | (x e. B /\ y = [_D / x]_C)})

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 950   = wceq 1099   e. wcel 1105  A.wral 1621  Vcvv 1786  [_csb 1972   (_ wss 2018  {copab 2634  dom cdm 3133  ran crn 3134   o. ccom 3137  Fun wfun 3139   Fn wfn 3140  ` cfv 3145

This theorem is referenced by:  oprcn 7859  kbass2t 10176  kbass5t 10179

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-rex 1626  df-v 1787  df-sbc 1913  df-csb 1973  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-uni 2472  df-br 2588  df-opab 2635  df-id 2797  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-fv 3161

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