Theorem fiv 10470

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 2086	. . . . . 6 |- (x = A -> (z (_ x <-> z (_ A))
2	1	3anbi1d 899	. . . . 5 |- (x = A -> ((z (_ x /\ z e. Fin /\ u = |^|z) <-> (z (_ A /\ z e. Fin /\ u = |^|z)))
3	2	exbidv 1281	. . . 4 |- (x = A -> (E.z(z (_ x /\ z e. Fin /\ u = |^|z) <-> E.z(z (_ A /\ z e. Fin /\ u = |^|z)))
4	3	abbidv 1580	. . 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 10469	. . . 4 |- fi = {<.x, y>. | y = {u | E.z(z (_ x /\ z e. Fin /\ u = |^|z)}}
6		relopab 3272	. . . . 5 |- Rel {<.x, y>. | y = {u | E.z(z (_ x /\ z e. Fin /\ u = |^|z)}}
7		resid 3406	. . . . 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 3401	. . . 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 1504	. . 3 |- fi = {<.x, y>. | (x e. V /\ y = {u | E.z(z (_ x /\ z e. Fin /\ u = |^|z)})}
11	4, 10	fvopab4g 3785	. 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 1820	. 2 |- (A e. B -> A e. V)
13		uniexg 2877	. . . . . 6 |- (A e. B -> U.A e. V)
14		pwexg 2752	. . . . . 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 2729	. . . . 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 1655	. . . 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 1556	. . 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 1816	. . . . . . . . 9 |- u e. V
22		eleq1 1537	. . . . . . . . . . . 12 |- (u = |^|z -> (u e. V <-> |^|z e. V))
23		intex 2734	. . . . . . . . . . . . 13 |- (z =/= (/) <-> |^|z e. V)
24		intssuni2 2560	. . . . . . . . . . . . . . . 16 |- ((z (_ A /\ z =/= (/)) -> |^|z (_ U.A)
25	24	ex 373	. . . . . . . . . . . . . . 15 |- (z (_ A -> (z =/= (/) -> |^|z (_ U.A))
26		sseq1 2085	. . . . . . . . . . . . . . . . 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 2408	. . . . . . . . . . . . . . . . . 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 829	. . . . . . 7 |- ((z (_ A /\ z e. Fin /\ u = |^|z) -> u e. P~U.A)
40	39	19.23aiv 1297	. . . . . 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 1578	. . 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 1556	. 2 |- (A e. B -> {u | E.z(z (_ A /\ z e. Fin /\ u = |^|z)} e. V)
45	11, 12, 44	sylanc 473	1 |- (A e. B -> (fi` A) = {u | E.z(z (_ A /\ z e. Fin /\ u = |^|z)})

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  E.wex 982  {cab 1466   =/= wne 1588  {crab 1651  Vcvv 1814   (_ wss 2050  (/)c0 2283  P~cpw 2405  U.cuni 2507  |^|cint 2537  {copab 2671   |` cres 3178  Rel wrel 3181  ` cfv 3188  Fincfn 4373  ficfi 10468

This theorem is referenced by:  fine2 10471  sppfi 10472  abfi2 10474

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-int 2538  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fv 3204  df-fi 10469

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