Theorem infi 10280

Description: The intersection of two finite intersections is a finite intersection. (Contributed by FL, 2-Sep-2008.)

Assertion

Ref	Expression
infi	|- (C e. D -> ((A e. ( fi ` C) /\ B e. ( fi ` C)) -> (A i^i B) e. ( fi ` C)))

Proof of Theorem infi

Step	Hyp	Ref	Expression
1		an6 1177	. . . . . . . . . . . . 13 |- (((x C_ C /\ x e. Fin /\ A = |^|x) /\ (y C_ C /\ y e. Fin /\ B = |^|y)) <-> ((x C_ C /\ y C_ C) /\ (x e. Fin /\ y e. Fin) /\ (A = |^|x /\ B = |^|y)))
2		unss 2780	. . . . . . . . . . . . . . 15 |- ((x C_ C /\ y C_ C) <-> (x u. y) C_ C)
3	2	biimpi 168	. . . . . . . . . . . . . 14 |- ((x C_ C /\ y C_ C) -> (x u. y) C_ C)
4		unfi 5644	. . . . . . . . . . . . . 14 |- ((x e. Fin /\ y e. Fin) -> (x u. y) e. Fin)
5		ineq12 2791	. . . . . . . . . . . . . . 15 |- ((A = |^|x /\ B = |^|y) -> (A i^i B) = (|^|x i^i |^|y))
6		intun 3249	. . . . . . . . . . . . . . 15 |- |^|(x u. y) = (|^|x i^i |^|y)
7	5, 6	syl6eqr 1946	. . . . . . . . . . . . . 14 |- ((A = |^|x /\ B = |^|y) -> (A i^i B) = |^|(x u. y))
8	3, 4, 7	3anim123i 1053	. . . . . . . . . . . . 13 |- (((x C_ C /\ y C_ C) /\ (x e. Fin /\ y e. Fin) /\ (A = |^|x /\ B = |^|y)) -> ((x u. y) C_ C /\ (x u. y) e. Fin /\ (A i^i B) = |^|(x u. y)))
9	1, 8	sylbi 216	. . . . . . . . . . . 12 |- (((x C_ C /\ x e. Fin /\ A = |^|x) /\ (y C_ C /\ y e. Fin /\ B = |^|y)) -> ((x u. y) C_ C /\ (x u. y) e. Fin /\ (A i^i B) = |^|(x u. y)))
10	9	ex 402	. . . . . . . . . . 11 |- ((x C_ C /\ x e. Fin /\ A = |^|x) -> ((y C_ C /\ y e. Fin /\ B = |^|y) -> ((x u. y) C_ C /\ (x u. y) e. Fin /\ (A i^i B) = |^|(x u. y))))
11		visset 2295	. . . . . . . . . . . . 13 |- x e. _V
12		visset 2295	. . . . . . . . . . . . 13 |- y e. _V
13	11, 12	unex 3796	. . . . . . . . . . . 12 |- (x u. y) e. _V
14		sseq1 2637	. . . . . . . . . . . . 13 |- (z = (x u. y) -> (z C_ C <-> (x u. y) C_ C))
15		eleq1 1957	. . . . . . . . . . . . 13 |- (z = (x u. y) -> (z e. Fin <-> (x u. y) e. Fin))
16		inteq 3217	. . . . . . . . . . . . . 14 |- (z = (x u. y) -> |^|z = |^|(x u. y))
17	16	eqeq2d 1895	. . . . . . . . . . . . 13 |- (z = (x u. y) -> ((A i^i B) = |^|z <-> (A i^i B) = |^|(x u. y)))
18	14, 15, 17	3anbi123d 1168	. . . . . . . . . . . 12 |- (z = (x u. y) -> ((z C_ C /\ z e. Fin /\ (A i^i B) = |^|z) <-> ((x u. y) C_ C /\ (x u. y) e. Fin /\ (A i^i B) = |^|(x u. y))))
19	13, 18	cla4ev 2371	. . . . . . . . . . 11 |- (((x u. y) C_ C /\ (x u. y) e. Fin /\ (A i^i B) = |^|(x u. y)) -> E.z(z C_ C /\ z e. Fin /\ (A i^i B) = |^|z))
20	10, 19	syl6com 64	. . . . . . . . . 10 |- ((y C_ C /\ y e. Fin /\ B = |^|y) -> ((x C_ C /\ x e. Fin /\ A = |^|x) -> E.z(z C_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
21	20	19.23aiv 1674	. . . . . . . . 9 |- (E.y(y C_ C /\ y e. Fin /\ B = |^|y) -> ((x C_ C /\ x e. Fin /\ A = |^|x) -> E.z(z C_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
22	21	com12 14	. . . . . . . 8 |- ((x C_ C /\ x e. Fin /\ A = |^|x) -> (E.y(y C_ C /\ y e. Fin /\ B = |^|y) -> E.z(z C_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
23	22	19.23aiv 1674	. . . . . . 7 |- (E.x(x C_ C /\ x e. Fin /\ A = |^|x) -> (E.y(y C_ C /\ y e. Fin /\ B = |^|y) -> E.z(z C_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
24	23	imp 377	. . . . . 6 |- ((E.x(x C_ C /\ x e. Fin /\ A = |^|x) /\ E.y(y C_ C /\ y e. Fin /\ B = |^|y)) -> E.z(z C_ C /\ z e. Fin /\ (A i^i B) = |^|z))
25	24	a1i 8	. . . . 5 |- (C e. D -> ((E.x(x C_ C /\ x e. Fin /\ A = |^|x) /\ E.y(y C_ C /\ y e. Fin /\ B = |^|y)) -> E.z(z C_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
26		sppfi 10218	. . . . . . . 8 |- ((A e. ( fi ` C) /\ C e. D) -> (A e. ( fi ` C) <-> E.x(x C_ C /\ x e. Fin /\ A = |^|x)))
27	26	biimpd 170	. . . . . . 7 |- ((A e. ( fi ` C) /\ C e. D) -> (A e. ( fi ` C) -> E.x(x C_ C /\ x e. Fin /\ A = |^|x)))
28	27	ex 402	. . . . . 6 |- (A e. ( fi ` C) -> (C e. D -> (A e. ( fi ` C) -> E.x(x C_ C /\ x e. Fin /\ A = |^|x))))
29	28	pm2.43b 81	. . . . 5 |- (C e. D -> (A e. ( fi ` C) -> E.x(x C_ C /\ x e. Fin /\ A = |^|x)))
30		sppfi 10218	. . . . . . . 8 |- ((B e. ( fi ` C) /\ C e. D) -> (B e. ( fi ` C) <-> E.y(y C_ C /\ y e. Fin /\ B = |^|y)))
31	30	biimpd 170	. . . . . . 7 |- ((B e. ( fi ` C) /\ C e. D) -> (B e. ( fi ` C) -> E.y(y C_ C /\ y e. Fin /\ B = |^|y)))
32	31	ex 402	. . . . . 6 |- (B e. ( fi ` C) -> (C e. D -> (B e. ( fi ` C) -> E.y(y C_ C /\ y e. Fin /\ B = |^|y))))
33	32	pm2.43b 81	. . . . 5 |- (C e. D -> (B e. ( fi ` C) -> E.y(y C_ C /\ y e. Fin /\ B = |^|y)))
34	25, 29, 33	syl2and 508	. . . 4 |- (C e. D -> ((A e. ( fi ` C) /\ B e. ( fi ` C)) -> E.z(z C_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
35	34	imp 377	. . 3 |- ((C e. D /\ (A e. ( fi ` C) /\ B e. ( fi ` C))) -> E.z(z C_ C /\ z e. Fin /\ (A i^i B) = |^|z))
36		inex1g 3454	. . . . . . 7 |- (A e. ( fi ` C) -> (A i^i B) e. _V)
37	36	adantr 425	. . . . . 6 |- ((A e. ( fi ` C) /\ B e. ( fi ` C)) -> (A i^i B) e. _V)
38	37	anim1i 361	. . . . 5 |- (((A e. ( fi ` C) /\ B e. ( fi ` C)) /\ C e. D) -> ((A i^i B) e. _V /\ C e. D))
39	38	ancoms 484	. . . 4 |- ((C e. D /\ (A e. ( fi ` C) /\ B e. ( fi ` C))) -> ((A i^i B) e. _V /\ C e. D))
40		sppfi 10218	. . . 4 |- (((A i^i B) e. _V /\ C e. D) -> ((A i^i B) e. ( fi ` C) <-> E.z(z C_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
41	39, 40	syl 12	. . 3 |- ((C e. D /\ (A e. ( fi ` C) /\ B e. ( fi ` C))) -> ((A i^i B) e. ( fi ` C) <-> E.z(z C_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
42	35, 41	mpbird 213	. 2 |- ((C e. D /\ (A e. ( fi ` C) /\ B e. ( fi ` C))) -> (A i^i B) e. ( fi ` C))
43	42	ex 402	1 |- (C e. D -> ((A e. ( fi ` C) /\ B e. ( fi ` C)) -> (A i^i B) e. ( fi ` C)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326  _Vcvv 2292   u. cun 2591   i^i cin 2592   C_ wss 2593  |^|cint 3214  ` cfv 3998  Fincfn 5426   fi cfi 10210

This theorem is referenced by:  fsubbas 10281  fgsb2 14925

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-rep 3428  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-3or 859  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-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  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-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-opr 4886  df-oprab 4887  df-rdg 5140  df-oadd 5179  df-er 5318  df-en 5427  df-fin 5430  df-fi 10211

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