Theorem fnopabco 15711

Description: Composition of a function with a function abstraction.

Hypotheses

Ref	Expression
fnopabco.1	|- (x e. A -> B e. C)
fnopabco.2	|- F = {<.x, y>. | (x e. A /\ y = B)}
fnopabco.3	|- G = {<.x, y>. | (x e. A /\ y = (H` B))}

Assertion

Ref	Expression
fnopabco	|- (H Fn C -> G = (H o. F))

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

Proof of Theorem fnopabco

Step	Hyp	Ref	Expression
1		eqfnfv 4766	. . 3 |- ((G Fn A /\ (H o. F) Fn A) -> (G = (H o. F) <-> (A = A /\ A.z e. A (G` z) = ((H o. F)` z))))
2		fvex 4689	. . . 4 |- (H` B) e. _V
3		fnopabco.3	. . . 4 |- G = {<.x, y>. | (x e. A /\ y = (H` B))}
4	2, 3	fnopab2 4549	. . 3 |- G Fn A
5		fnopabco.2	. . . . . 6 |- F = {<.x, y>. | (x e. A /\ y = B)}
6		fnopabco.1	. . . . . 6 |- (x e. A -> B e. C)
7	5, 6	fopab 4800	. . . . 5 |- F:A-->C
8		ffn 4562	. . . . 5 |- (F:A-->C -> F Fn A)
9	7, 8	ax-mp 7	. . . 4 |- F Fn A
10		frn 4569	. . . . 5 |- (F:A-->C -> ran F C_ C)
11	7, 10	ax-mp 7	. . . 4 |- ran F C_ C
12		fnco 4521	. . . 4 |- ((H Fn C /\ F Fn A /\ ran F C_ C) -> (H o. F) Fn A)
13	9, 11, 12	mp3an23 1183	. . 3 |- (H Fn C -> (H o. F) Fn A)
14	1, 4, 13	sylancr 526	. 2 |- (H Fn C -> (G = (H o. F) <-> (A = A /\ A.z e. A (G` z) = ((H o. F)` z))))
15		eqidd 1885	. 2 |- (H Fn C -> A = A)
16	6	sbcth2 2514	. . . . . . . 8 |- (z e. A -> [z / x]B e. C)
17		visset 2295	. . . . . . . . 9 |- z e. _V
18		sbcel1g 2556	. . . . . . . . 9 |- (z e. _V -> ([z / x]B e. C <-> [_z / x]_B e. C))
19	17, 18	ax-mp 7	. . . . . . . 8 |- ([z / x]B e. C <-> [_z / x]_B e. C)
20	16, 19	sylib 215	. . . . . . 7 |- (z e. A -> [_z / x]_B e. C)
21		ax-17 1317	. . . . . . . 8 |- (y e. z -> A.x y e. z)
22	17, 21	hbcsb1 2568	. . . . . . . 8 |- (y e. [_z / x]_B -> A.x y e. [_z / x]_B)
23		csbeq1a 2546	. . . . . . . 8 |- (x = z -> B = [_z / x]_B)
24	21, 22, 23, 5	fvopab4gf 4744	. . . . . . 7 |- ((z e. A /\ [_z / x]_B e. C) -> (F` z) = [_z / x]_B)
25	20, 24	mpdan 768	. . . . . 6 |- (z e. A -> (F` z) = [_z / x]_B)
26	25	adantl 424	. . . . 5 |- ((H Fn C /\ z e. A) -> (F` z) = [_z / x]_B)
27	26	fveq2d 4685	. . . 4 |- ((H Fn C /\ z e. A) -> (H` (F` z)) = (H` [_z / x]_B))
28		fnfun 4510	. . . . . 6 |- (H Fn C -> Fun H)
29	28	adantr 425	. . . . 5 |- ((H Fn C /\ z e. A) -> Fun H)
30		ffun 4565	. . . . . . 7 |- (F:A-->C -> Fun F)
31	7, 30	ax-mp 7	. . . . . 6 |- Fun F
32	31	a1i 8	. . . . 5 |- ((H Fn C /\ z e. A) -> Fun F)
33	7	fdmi 4568	. . . . . . . 8 |- dom F = A
34	33	eleq2i 1961	. . . . . . 7 |- (z e. dom F <-> z e. A)
35	34	biimpri 169	. . . . . 6 |- (z e. A -> z e. dom F)
36	35	adantl 424	. . . . 5 |- ((H Fn C /\ z e. A) -> z e. dom F)
37		fvco 4736	. . . . 5 |- ((Fun H /\ Fun F /\ z e. dom F) -> ((H o. F)` z) = (H` (F` z)))
38	29, 32, 36, 37	syl111anc 1100	. . . 4 |- ((H Fn C /\ z e. A) -> ((H o. F)` z) = (H` (F` z)))
39		fvex 4689	. . . . . 6 |- (H` [_z / x]_B) e. _V
40		ax-17 1317	. . . . . . . 8 |- (y e. H -> A.x y e. H)
41	40, 22	hbfv 4686	. . . . . . 7 |- (y e. (H` [_z / x]_B) -> A.x y e. (H` [_z / x]_B))
42	23	fveq2d 4685	. . . . . . 7 |- (x = z -> (H` B) = (H` [_z / x]_B))
43	21, 41, 42, 3	fvopab4gf 4744	. . . . . 6 |- ((z e. A /\ (H` [_z / x]_B) e. _V) -> (G` z) = (H` [_z / x]_B))
44	39, 43	mpan2 760	. . . . 5 |- (z e. A -> (G` z) = (H` [_z / x]_B))
45	44	adantl 424	. . . 4 |- ((H Fn C /\ z e. A) -> (G` z) = (H` [_z / x]_B))
46	27, 38, 45	3eqtr4rd 1939	. . 3 |- ((H Fn C /\ z e. A) -> (G` z) = ((H o. F)` z))
47	46	r19.21aiva 2176	. 2 |- (H Fn C -> A.z e. A (G` z) = ((H o. F)` z))
48	14, 15, 47	mpbir2and 802	1 |- (H Fn C -> G = (H o. F))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  [wsbc 1534  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  -->wf 3994  ` cfv 3998

This theorem is referenced by:  opropabco 15712  pcopt 16084  pcoass 16085  pcorev 16087  pi1gp 16095

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-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-f 4010  df-fv 4014

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