Theorem fiv 10482

Description: The set of all the finite intersections of the elements of A.

Assertion

Ref	Expression
fiv	|- (A e. B -> (fi` A) = {u | E.z(z (_ A /\ z e. Fin /\ u = |^|z)})

Distinct variable group:   u,A,z

Proof of Theorem fiv

Step	Hyp	Ref	Expression
1		sseq2 2083	. . . . . 6 |- (x = A -> (z (_ x <-> z (_ A))
2	1	3anbi1d 897	. . . . 5 |- (x = A -> ((z (_ x /\ z e. Fin /\ u = |^|z) <-> (z (_ A /\ z e. Fin /\ u = |^|z)))
3	2	exbidv 1279	. . . 4 |- (x = A -> (E.z(z (_ x /\ z e. Fin /\ u = |^|z) <-> E.z(z (_ A /\ z e. Fin /\ u = |^|z)))
4	3	abbidv 1577	. . 3 |- (x = A -> {u | E.z(z (_ x /\ z e. Fin /\ u = |^|z)} = {u | E.z(z (_ A /\ z e. Fin /\ u = |^|z)})
5		df-fi 10480	. . . 4 |- fi = {<.x, y>. | y = {u | E.z(z (_ x /\ z e. Fin /\ u = |^|z)}}
6		relopab 3266	. . . . 5 |- Rel {<.x, y>. | y = {u | E.z(z (_ x /\ z e. Fin /\ u = |^|z)}}
7		resid 3400	. . . . 5 |- (Rel {<.x, y>. | y = {u | E.z(z (_ x /\ z e. Fin /\ u = |^|z)}} -> ({<.x, y>. | y = {u | E.z(z (_ x /\ z e. Fin /\ u = |^|z)}} |` V) = {<.x, y>. | y = {u | E.z(z (_ x /\ z e. Fin /\ u = |^|z)}})
8	6, 7	ax-mp 7	. . . 4 |- ({<.x, y>. | y = {u | E.z(z (_ x /\ z e. Fin /\ u = |^|z)}} |` V) = {<.x, y>. | y = {u | E.z(z (_ x /\ z e. Fin /\ u = |^|z)}}
9		resopab 3395	. . . 4 |- ({<.x, y>. | y = {u | E.z(z (_ x /\ z e. Fin /\ u = |^|z)}} |` V) = {<.x, y>. | (x e. V /\ y = {u | E.z(z (_ x /\ z e. Fin /\ u = |^|z)})}
10	5, 8, 9	3eqtr2 1501	. . 3 |- fi = {<.x, y>. | (x e. V /\ y = {u | E.z(z (_ x /\ z e. Fin /\ u = |^|z)})}
11	4, 10	fvopab4g 3779	. 2 |- ((A e. V /\ {u | E.z(z (_ A /\ z e. Fin /\ u = |^|z)} e. V) -> (fi` A) = {u | E.z(z (_ A /\ z e. Fin /\ u = |^|z)})
12		elisset 1817	. 2 |- (A e. B -> A e. V)
13		uniexg 2871	. . . . . 6 |- (A e. B -> U.A e. V)
14		pwexg 2746	. . . . . 6 |- (U.A e. V -> P~U.A e. V)
15	13, 14	syl 10	. . . . 5 |- (A e. B -> P~U.A e. V)
16		rabexg 2724	. . . . 5 |- (P~U.A e. V -> {u e. P~U.A | E.z(z (_ A /\ z e. Fin /\ u = |^|z)} e. V)
17	15, 16	syl 10	. . . 4 |- (A e. B -> {u e. P~U.A | E.z(z (_ A /\ z e. Fin /\ u = |^|z)} e. V)
18		df-rab 1652	. . . 4 |- {u e. P~U.A | E.z(z (_ A /\ z e. Fin /\ u = |^|z)} = {u | (u e. P~U.A /\ E.z(z (_ A /\ z e. Fin /\ u = |^|z))}
19	17, 18	syl5eqelr 1553	. . 3 |- (A e. B -> {u | (u e. P~U.A /\ E.z(z (_ A /\ z e. Fin /\ u = |^|z))} e. V)
20		pm3.27 323	. . . . 5 |- ((u e. P~U.A /\ E.z(z (_ A /\ z e. Fin /\ u = |^|z)) -> E.z(z (_ A /\ z e. Fin /\ u = |^|z))
21		visset 1813	. . . . . . . . 9 |- u e. V
22		eleq1 1534	. . . . . . . . . . . 12 |- (u = |^|z -> (u e. V <-> |^|z e. V))
23		intex 2729	. . . . . . . . . . . . 13 |- (z =/= (/) <-> |^|z e. V)
24		intssuni2 2556	. . . . . . . . . . . . . . . 16 |- ((z (_ A /\ z =/= (/)) -> |^|z (_ U.A)
25	24	ex 373	. . . . . . . . . . . . . . 15 |- (z (_ A -> (z =/= (/) -> |^|z (_ U.A))
26		sseq1 2082	. . . . . . . . . . . . . . . . 17 |- (u = |^|z -> (u (_ U.A <-> |^|z (_ U.A))
27	26	biimprd 154	. . . . . . . . . . . . . . . 16 |- (u = |^|z -> (|^|z (_ U.A -> u (_ U.A))
28	21	elpw 2404	. . . . . . . . . . . . . . . . . 18 |- (u e. P~U.A <-> u (_ U.A)
29	28	biimpr 152	. . . . . . . . . . . . . . . . 17 |- (u (_ U.A -> u e. P~U.A)
30	29	a1d 12	. . . . . . . . . . . . . . . 16 |- (u (_ U.A -> (z e. Fin -> u e. P~U.A))
31	27, 30	syl6com 53	. . . . . . . . . . . . . . 15 |- (|^|z (_ U.A -> (u = |^|z -> (z e. Fin -> u e. P~U.A)))
32	25, 31	syl6 22	. . . . . . . . . . . . . 14 |- (z (_ A -> (z =/= (/) -> (u = |^|z -> (z e. Fin -> u e. P~U.A))))
33	32	com3l 34	. . . . . . . . . . . . 13 |- (z =/= (/) -> (u = |^|z -> (z (_ A -> (z e. Fin -> u e. P~U.A))))
34	23, 33	sylbir 201	. . . . . . . . . . . 12 |- (|^|z e. V -> (u = |^|z -> (z (_ A -> (z e. Fin -> u e. P~U.A))))
35	22, 34	syl6bi 214	. . . . . . . . . . 11 |- (u = |^|z -> (u e. V -> (u = |^|z -> (z (_ A -> (z e. Fin -> u e. P~U.A)))))
36	35	pm2.43a 66	. . . . . . . . . 10 |- (u = |^|z -> (u e. V -> (z (_ A -> (z e. Fin -> u e. P~U.A))))
37	36	com4l 39	. . . . . . . . 9 |- (u e. V -> (z (_ A -> (z e. Fin -> (u = |^|z -> u e. P~U.A))))
38	21, 37	ax-mp 7	. . . . . . . 8 |- (z (_ A -> (z e. Fin -> (u = |^|z -> u e. P~U.A)))
39	38	3imp 827	. . . . . . 7 |- ((z (_ A /\ z e. Fin /\ u = |^|z) -> u e. P~U.A)
40	39	19.23aiv 1295	. . . . . 6 |- (E.z(z (_ A /\ z e. Fin /\ u = |^|z) -> u e. P~U.A)
41	40	ancri 297	. . . . 5 |- (E.z(z (_ A /\ z e. Fin /\ u = |^|z) -> (u e. P~U.A /\ E.z(z (_ A /\ z e. Fin /\ u = |^|z)))
42	20, 41	impbi 157	. . . 4 |- ((u e. P~U.A /\ E.z(z (_ A /\ z e. Fin /\ u = |^|z)) <-> E.z(z (_ A /\ z e. Fin /\ u = |^|z))
43	42	abbii 1575	. . 3 |- {u | (u e. P~U.A /\ E.z(z (_ A /\ z e. Fin /\ u = |^|z))} = {u | E.z(z (_ A /\ z e. Fin /\ u = |^|z)}
44	19, 43	syl5eqelr 1553	. 2 |- (A e. B -> {u | E.z(z (_ A /\ z e. Fin /\ u = |^|z)} e. V)
45	11, 12, 44	sylanc 471	1 |- (A e. B -> (fi` A) = {u | E.z(z (_ A /\ z e. Fin /\ u = |^|z)})

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wex 980  {cab 1463   =/= wne 1585  {crab 1648  Vcvv 1811   (_ wss 2047  (/)c0 2280  P~cpw 2401  U.cuni 2503  |^|cint 2533  {copab 2666   |` cres 3172  Rel wrel 3175  ` cfv 3182  Fincfn 4367  ficfi 10479

This theorem is referenced by:  fine2 10484  sppfi 10486  abfi2 10490

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-int 2534  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-fi 10480

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