Theorem fopabco 3827

Description: Composition of two functions expressed as ordered-pair class abstractions. Note that v may be assigned to w, y, or z if desired.

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 (_ B -> (G o. F) = H)

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

Proof of Theorem fopabco

Step	Hyp	Ref	Expression
1		fopabco.5	. . . . . . 7 |- F = {<.x, y>. | (x e. A /\ y = R)}
2		hbopab1 2809	. . . . . . 7 |- (u e. {<.x, y>. | (x e. A /\ y = R)} -> A.x u e. {<.x, y>. | (x e. A /\ y = R)})
3	1, 2	hbxfr 1561	. . . . . 6 |- (u e. F -> A.x u e. F)
4	3	hbrn 3347	. . . . 5 |- (u e. ran F -> A.x u e. ran F)
5		ax-17 970	. . . . 5 |- (u e. B -> A.x u e. B)
6	4, 5	hbss 2059	. . . 4 |- (ran F (_ B -> A.xran F (_ B)
7		fopabco.1	. . . . . . . . . . 11 |- R e. V
8		fvopab2 3786	. . . . . . . . . . 11 |- ((x e. A /\ R e. V) -> ({<.x, y>. | (x e. A /\ y = R)}` x) = R)
9	7, 8	mpan2 695	. . . . . . . . . 10 |- (x e. A -> ({<.x, y>. | (x e. A /\ y = R)}` x) = R)
10	1	fveq1i 3720	. . . . . . . . . 10 |- (F` x) = ({<.x, y>. | (x e. A /\ y = R)}` x)
11	9, 10	syl5eq 1517	. . . . . . . . 9 |- (x e. A -> (F` x) = R)
12	11	fveq2d 3723	. . . . . . . 8 |- (x e. A -> (G` (F` x)) = (G` R))
13	12	adantl 388	. . . . . . 7 |- ((ran F (_ B /\ x e. A) -> (G` (F` x)) = (G` R))
14		ffvelrn 3809	. . . . . . . . . 10 |- ((F:A-->B /\ x e. A) -> (F` x) e. B)
15	7, 1	fnopab2 3614	. . . . . . . . . . 11 |- F Fn A
16		df-f 3190	. . . . . . . . . . . 12 |- (F:A-->B <-> (F Fn A /\ ran F (_ B))
17	16	biimpr 152	. . . . . . . . . . 11 |- ((F Fn A /\ ran F (_ B) -> F:A-->B)
18	15, 17	mpan 694	. . . . . . . . . 10 |- (ran F (_ B -> F:A-->B)
19	14, 18	sylan 448	. . . . . . . . 9 |- ((ran F (_ B /\ x e. A) -> (F` x) e. B)
20	11	eleq1d 1538	. . . . . . . . . 10 |- (x e. A -> ((F` x) e. B <-> R e. B))
21	20	adantl 388	. . . . . . . . 9 |- ((ran F (_ B /\ x e. A) -> ((F` x) e. B <-> R e. B))
22	19, 21	mpbid 195	. . . . . . . 8 |- ((ran F (_ B /\ x e. A) -> R e. B)
23		fopabco.4	. . . . . . . . 9 |- (z = R -> S = T)
24		fopabco.6	. . . . . . . . 9 |- G = {<.z, w>. | (z e. B /\ w = S)}
25		fopabco.3	. . . . . . . . 9 |- T e. V
26	23, 24, 25	fvopab4 3775	. . . . . . . 8 |- (R e. B -> (G` R) = T)
27	22, 26	syl 10	. . . . . . 7 |- ((ran F (_ B /\ x e. A) -> (G` R) = T)
28	13, 27	eqtrd 1505	. . . . . 6 |- ((ran F (_ B /\ x e. A) -> (G` (F` x)) = T)
29	7, 1	dmopab2 3615	. . . . . . . . 9 |- dom F = A
30	29	eleq2i 1536	. . . . . . . 8 |- (x e. dom F <-> x e. A)
31		fopabco.2	. . . . . . . . . . 11 |- S e. V
32	31, 24	fnopab2 3614	. . . . . . . . . 10 |- G Fn B
33		fnfun 3581	. . . . . . . . . 10 |- (G Fn B -> Fun G)
34	32, 33	ax-mp 7	. . . . . . . . 9 |- Fun G
35		fnfun 3581	. . . . . . . . . 10 |- (F Fn A -> Fun F)
36	15, 35	ax-mp 7	. . . . . . . . 9 |- Fun F
37		fvco 3769	. . . . . . . . 9 |- ((Fun G /\ Fun F /\ x e. dom F) -> ((G o. F)` x) = (G` (F` x)))
38	34, 36, 37	mp3an12 905	. . . . . . . 8 |- (x e. dom F -> ((G o. F)` x) = (G` (F` x)))
39	30, 38	sylbir 201	. . . . . . 7 |- (x e. A -> ((G o. F)` x) = (G` (F` x)))
40	39	adantl 388	. . . . . 6 |- ((ran F (_ B /\ x e. A) -> ((G o. F)` x) = (G` (F` x)))
41		fvopab2 3786	. . . . . . . . 9 |- ((x e. A /\ T e. V) -> ({<.x, v>. | (x e. A /\ v = T)}` x) = T)
42		fopabco.7	. . . . . . . . . 10 |- H = {<.x, v>. | (x e. A /\ v = T)}
43	42	fveq1i 3720	. . . . . . . . 9 |- (H` x) = ({<.x, v>. | (x e. A /\ v = T)}` x)
44	41, 43	syl5eq 1517	. . . . . . . 8 |- ((x e. A /\ T e. V) -> (H` x) = T)
45	25, 44	mpan2 695	. . . . . . 7 |- (x e. A -> (H` x) = T)
46	45	adantl 388	. . . . . 6 |- ((ran F (_ B /\ x e. A) -> (H` x) = T)
47	28, 40, 46	3eqtr4d 1515	. . . . 5 |- ((ran F (_ B /\ x e. A) -> ((G o. F)` x) = (H` x))
48	47	ex 373	. . . 4 |- (ran F (_ B -> (x e. A -> ((G o. F)` x) = (H` x)))
49	6, 48	r19.21ai 1710	. . 3 |- (ran F (_ B -> A.x e. A ((G o. F)` x) = (H` x))
50		eqid 1474	. . 3 |- A = A
51	49, 50	jctil 292	. 2 |- (ran F (_ B -> (A = A /\ A.x e. A ((G o. F)` x) = (H` x)))
52		fnco 3591	. . . 4 |- ((G Fn B /\ F Fn A /\ ran F (_ B) -> (G o. F) Fn A)
53	32, 15, 52	mp3an12 905	. . 3 |- (ran F (_ B -> (G o. F) Fn A)
54	25, 42	fnopab2 3614	. . . 4 |- H Fn A
55		ax-17 970	. . . . . 6 |- (u e. G -> A.x u e. G)
56	55, 3	hbco 3283	. . . . 5 |- (u e. (G o. F) -> A.x u e. (G o. F))
57		hbopab1 2809	. . . . . 6 |- (u e. {<.x, v>. | (x e. A /\ v = T)} -> A.x u e. {<.x, v>. | (x e. A /\ v = T)})
58	42, 57	hbxfr 1561	. . . . 5 |- (u e. H -> A.x u e. H)
59	56, 58	eqfnfvf 3793	. . . 4 |- (((G o. F) Fn A /\ H Fn A) -> ((G o. F) = H <-> (A = A /\ A.x e. A ((G o. F)` x) = (H` x))))
60	54, 59	mpan2 695	. . 3 |- ((G o. F) Fn A -> ((G o. F) = H <-> (A = A /\ A.x e. A ((G o. F)` x) = (H` x))))
61	53, 60	syl 10	. 2 |- (ran F (_ B -> ((G o. F) = H <-> (A = A /\ A.x e. A ((G o. F)` x) = (H` x))))
62	51, 61	mpbird 196	1 |- (ran F (_ B -> (G o. F) = H)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 955   e. wcel 957  A.wral 1643  Vcvv 1808   (_ wss 2044  {copab 2662  dom cdm 3166  ran crn 3167   o. ccom 3170  Fun wfun 3172   Fn wfn 3173  -->wf 3174  ` cfv 3178

This theorem is referenced by:  ip1cnilem2 8336  ip1cnilem3 8337  ipasslem6 8454

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1209  ax-11o 1217  ax-ext 1458  ax-sep 2699  ax-pow 2738  ax-pr 2775  ax-un 2862

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1171  df-eu 1381  df-mo 1382  df-clab 1463  df-cleq 1468  df-clel 1471  df-ne 1585  df-ral 1647  df-rex 1648  df-v 1809  df-dif 2046  df-un 2047  df-in 2048  df-ss 2050  df-nul 2278  df-pw 2399  df-sn 2409  df-pr 2410  df-op 2413  df-uni 2500  df-br 2616  df-opab 2663  df-id 2831  df-xp 3180  df-rel 3181  df-cnv 3182  df-co 3183  df-dm 3184  df-rn 3185  df-res 3186  df-ima 3187  df-fun 3188  df-fn 3189  df-f 3190  df-fv 3194

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