Theorem fopabco 4805

Description: Composition of two functions expressed as ordered-pair class abstractions. Note that v may be assigned to w, y, or z if desired. (Unnecessary distinct variable restrictions were removed by David Abernethy, 21-Jan-2012.)

Hypotheses

Ref	Expression
fopabco.1	|- R e. _V
fopabco.2	|- S e. _V
fopabco.3	|- T e. _V
fopabco.4	|- (z = R -> S = T)
fopabco.5	|- F = {<.x, y>. | (x e. A /\ y = R)}
fopabco.6	|- G = {<.z, w>. | (z e. B /\ w = S)}
fopabco.7	|- H = {<.x, v>. | (x e. A /\ v = T)}

Assertion

Ref	Expression
fopabco	|- (ran F C_ B -> (G o. F) = H)

Distinct variable groups:   x,y,A   x,B   z,w,B   x,G   w,S   y,R   w,R,z   w,T,z   x,v,A   v,T

Proof of Theorem fopabco

Step	Hyp	Ref	Expression
1		fopabco.5	. . . . . 6 |- F = {<.x, y>. | (x e. A /\ y = R)}
2		hbopab1 3562	. . . . . 6 |- (u e. {<.x, y>. | (x e. A /\ y = R)} -> A.x u e. {<.x, y>. | (x e. A /\ y = R)})
3	1, 2	hbxfr 1992	. . . . 5 |- (u e. F -> A.x u e. F)
4	3	hbrn 4198	. . . 4 |- (u e. ran F -> A.x u e. ran F)
5		ax-17 1317	. . . 4 |- (u e. B -> A.x u e. B)
6	4, 5	hbss 2614	. . 3 |- (ran F C_ B -> A.xran F C_ B)
7		fopabco.1	. . . . . . . . . 10 |- R e. _V
8		fvopab2 4754	. . . . . . . . . 10 |- ((x e. A /\ R e. _V) -> ({<.x, y>. | (x e. A /\ y = R)}` x) = R)
9	7, 8	mpan2 760	. . . . . . . . 9 |- (x e. A -> ({<.x, y>. | (x e. A /\ y = R)}` x) = R)
10	1	fveq1i 4682	. . . . . . . . 9 |- (F` x) = ({<.x, y>. | (x e. A /\ y = R)}` x)
11	9, 10	syl5eq 1940	. . . . . . . 8 |- (x e. A -> (F` x) = R)
12	11	fveq2d 4685	. . . . . . 7 |- (x e. A -> (G` (F` x)) = (G` R))
13	12	adantl 424	. . . . . 6 |- ((ran F C_ B /\ x e. A) -> (G` (F` x)) = (G` R))
14		ffvelrn 4787	. . . . . . . . 9 |- ((F:A-->B /\ x e. A) -> (F` x) e. B)
15	7, 1	fnopab2 4549	. . . . . . . . . 10 |- F Fn A
16		df-f 4010	. . . . . . . . . . 11 |- (F:A-->B <-> (F Fn A /\ ran F C_ B))
17	16	biimpri 169	. . . . . . . . . 10 |- ((F Fn A /\ ran F C_ B) -> F:A-->B)
18	15, 17	mpan 759	. . . . . . . . 9 |- (ran F C_ B -> F:A-->B)
19	14, 18	sylan 497	. . . . . . . 8 |- ((ran F C_ B /\ x e. A) -> (F` x) e. B)
20	11	eleq1d 1963	. . . . . . . . 9 |- (x e. A -> ((F` x) e. B <-> R e. B))
21	20	adantl 424	. . . . . . . 8 |- ((ran F C_ B /\ x e. A) -> ((F` x) e. B <-> R e. B))
22	19, 21	mpbid 212	. . . . . . 7 |- ((ran F C_ B /\ x e. A) -> R e. B)
23		fopabco.4	. . . . . . . 8 |- (z = R -> S = T)
24		fopabco.6	. . . . . . . 8 |- G = {<.z, w>. | (z e. B /\ w = S)}
25		fopabco.3	. . . . . . . 8 |- T e. _V
26	23, 24, 25	fvopab4 4743	. . . . . . 7 |- (R e. B -> (G` R) = T)
27	22, 26	syl 12	. . . . . 6 |- ((ran F C_ B /\ x e. A) -> (G` R) = T)
28	13, 27	eqtrd 1925	. . . . 5 |- ((ran F C_ B /\ x e. A) -> (G` (F` x)) = T)
29	7, 1	dmopab2 4550	. . . . . . . 8 |- dom F = A
30	29	eleq2i 1961	. . . . . . 7 |- (x e. dom F <-> x e. A)
31		fopabco.2	. . . . . . . . . 10 |- S e. _V
32	31, 24	fnopab2 4549	. . . . . . . . 9 |- G Fn B
33		fnfun 4510	. . . . . . . . 9 |- (G Fn B -> Fun G)
34	32, 33	ax-mp 7	. . . . . . . 8 |- Fun G
35		fnfun 4510	. . . . . . . . 9 |- (F Fn A -> Fun F)
36	15, 35	ax-mp 7	. . . . . . . 8 |- Fun F
37		fvco 4736	. . . . . . . 8 |- ((Fun G /\ Fun F /\ x e. dom F) -> ((G o. F)` x) = (G` (F` x)))
38	34, 36, 37	mp3an12 1181	. . . . . . 7 |- (x e. dom F -> ((G o. F)` x) = (G` (F` x)))
39	30, 38	sylbir 218	. . . . . 6 |- (x e. A -> ((G o. F)` x) = (G` (F` x)))
40	39	adantl 424	. . . . 5 |- ((ran F C_ B /\ x e. A) -> ((G o. F)` x) = (G` (F` x)))
41		fvopab2 4754	. . . . . . . 8 |- ((x e. A /\ T e. _V) -> ({<.x, v>. | (x e. A /\ v = T)}` x) = T)
42		fopabco.7	. . . . . . . . 9 |- H = {<.x, v>. | (x e. A /\ v = T)}
43	42	fveq1i 4682	. . . . . . . 8 |- (H` x) = ({<.x, v>. | (x e. A /\ v = T)}` x)
44	41, 43	syl5eq 1940	. . . . . . 7 |- ((x e. A /\ T e. _V) -> (H` x) = T)
45	25, 44	mpan2 760	. . . . . 6 |- (x e. A -> (H` x) = T)
46	45	adantl 424	. . . . 5 |- ((ran F C_ B /\ x e. A) -> (H` x) = T)
47	28, 40, 46	3eqtr4d 1937	. . . 4 |- ((ran F C_ B /\ x e. A) -> ((G o. F)` x) = (H` x))
48	47	ex 402	. . 3 |- (ran F C_ B -> (x e. A -> ((G o. F)` x) = (H` x)))
49	6, 48	r19.21ai 2174	. 2 |- (ran F C_ B -> A.x e. A ((G o. F)` x) = (H` x))
50		ax-17 1317	. . . . 5 |- (u e. G -> A.x u e. G)
51	50, 3	hbco 4129	. . . 4 |- (u e. (G o. F) -> A.x u e. (G o. F))
52		hbopab1 3562	. . . . 5 |- (u e. {<.x, v>. | (x e. A /\ v = T)} -> A.x u e. {<.x, v>. | (x e. A /\ v = T)})
53	42, 52	hbxfr 1992	. . . 4 |- (u e. H -> A.x u e. H)
54	51, 53	eqfnfv2f 4770	. . 3 |- (((G o. F) Fn A /\ H Fn A) -> ((G o. F) = H <-> A.x e. A ((G o. F)` x) = (H` x)))
55		fnco 4521	. . . 4 |- ((G Fn B /\ F Fn A /\ ran F C_ B) -> (G o. F) Fn A)
56	32, 15, 55	mp3an12 1181	. . 3 |- (ran F C_ B -> (G o. F) Fn A)
57	25, 42	fnopab2 4549	. . 3 |- H Fn A
58	54, 56, 57	sylancl 525	. 2 |- (ran F C_ B -> ((G o. F) = H <-> A.x e. A ((G o. F)` x) = (H` x)))
59	49, 58	mpbird 213	1 |- (ran F C_ B -> (G o. F) = H)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  {copab 3395  dom cdm 3986  ran crn 3987   o. ccom 3990  Fun wfun 3992   Fn wfn 3993  -->wf 3994  ` cfv 3998

This theorem is referenced by:  ip1cnilem2 9713  ip1cnilem3 9714  ipasslem6 9836  cmprtr 14760  pcohtpylem3 16082  pcopt 16084  pcoass 16085

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-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

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