Theorem infi 10559

Description: The intersection of two finite intersections is a finite intersection.

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 904	. . . . . . . . . . . . 13 |- (((x (_ C /\ x e. Fin /\ A = |^|x) /\ (y (_ C /\ y e. Fin /\ B = |^|y)) <-> ((x (_ C /\ y (_ C) /\ (x e. Fin /\ y e. Fin) /\ (A = |^|x /\ B = |^|y)))
2		unss 2207	. . . . . . . . . . . . . . 15 |- ((x (_ C /\ y (_ C) <-> (x u. y) (_ C)
3	2	biimp 151	. . . . . . . . . . . . . 14 |- ((x (_ C /\ y (_ C) -> (x u. y) (_ C)
4		unfi 4563	. . . . . . . . . . . . . 14 |- ((x e. Fin /\ y e. Fin) -> (x u. y) e. Fin)
5		ineq12 2215	. . . . . . . . . . . . . . 15 |- ((A = |^|x /\ B = |^|y) -> (A i^i B) = (|^|x i^i |^|y))
6		intun 2566	. . . . . . . . . . . . . . 15 |- |^|(x u. y) = (|^|x i^i |^|y)
7	5, 6	syl6eqr 1528	. . . . . . . . . . . . . 14 |- ((A = |^|x /\ B = |^|y) -> (A i^i B) = |^|(x u. y))
8	3, 4, 7	3anim123i 823	. . . . . . . . . . . . 13 |- (((x (_ C /\ y (_ C) /\ (x e. Fin /\ y e. Fin) /\ (A = |^|x /\ B = |^|y)) -> ((x u. y) (_ C /\ (x u. y) e. Fin /\ (A i^i B) = |^|(x u. y)))
9	1, 8	sylbi 199	. . . . . . . . . . . 12 |- (((x (_ C /\ x e. Fin /\ A = |^|x) /\ (y (_ C /\ y e. Fin /\ B = |^|y)) -> ((x u. y) (_ C /\ (x u. y) e. Fin /\ (A i^i B) = |^|(x u. y)))
10	9	ex 373	. . . . . . . . . . 11 |- ((x (_ C /\ x e. Fin /\ A = |^|x) -> ((y (_ C /\ y e. Fin /\ B = |^|y) -> ((x u. y) (_ C /\ (x u. y) e. Fin /\ (A i^i B) = |^|(x u. y))))
11		visset 1816	. . . . . . . . . . . . 13 |- x e. V
12		visset 1816	. . . . . . . . . . . . 13 |- y e. V
13	11, 12	unex 2878	. . . . . . . . . . . 12 |- (x u. y) e. V
14		sseq1 2085	. . . . . . . . . . . . 13 |- (z = (x u. y) -> (z (_ C <-> (x u. y) (_ C))
15		eleq1 1537	. . . . . . . . . . . . 13 |- (z = (x u. y) -> (z e. Fin <-> (x u. y) e. Fin))
16		inteq 2540	. . . . . . . . . . . . . 14 |- (z = (x u. y) -> |^|z = |^|(x u. y))
17	16	eqeq2d 1489	. . . . . . . . . . . . 13 |- (z = (x u. y) -> ((A i^i B) = |^|z <-> (A i^i B) = |^|(x u. y)))
18	14, 15, 17	3anbi123d 895	. . . . . . . . . . . 12 |- (z = (x u. y) -> ((z (_ C /\ z e. Fin /\ (A i^i B) = |^|z) <-> ((x u. y) (_ C /\ (x u. y) e. Fin /\ (A i^i B) = |^|(x u. y))))
19	13, 18	cla4ev 1872	. . . . . . . . . . 11 |- (((x u. y) (_ C /\ (x u. y) e. Fin /\ (A i^i B) = |^|(x u. y)) -> E.z(z (_ C /\ z e. Fin /\ (A i^i B) = |^|z))
20	10, 19	syl6com 53	. . . . . . . . . 10 |- ((y (_ C /\ y e. Fin /\ B = |^|y) -> ((x (_ C /\ x e. Fin /\ A = |^|x) -> E.z(z (_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
21	20	19.23aiv 1297	. . . . . . . . 9 |- (E.y(y (_ C /\ y e. Fin /\ B = |^|y) -> ((x (_ C /\ x e. Fin /\ A = |^|x) -> E.z(z (_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
22	21	com12 11	. . . . . . . 8 |- ((x (_ C /\ x e. Fin /\ A = |^|x) -> (E.y(y (_ C /\ y e. Fin /\ B = |^|y) -> E.z(z (_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
23	22	19.23aiv 1297	. . . . . . 7 |- (E.x(x (_ C /\ x e. Fin /\ A = |^|x) -> (E.y(y (_ C /\ y e. Fin /\ B = |^|y) -> E.z(z (_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
24	23	imp 350	. . . . . 6 |- ((E.x(x (_ C /\ x e. Fin /\ A = |^|x) /\ E.y(y (_ C /\ y e. Fin /\ B = |^|y)) -> E.z(z (_ C /\ z e. Fin /\ (A i^i B) = |^|z))
25	24	a1i 8	. . . . 5 |- (C e. D -> ((E.x(x (_ C /\ x e. Fin /\ A = |^|x) /\ E.y(y (_ C /\ y e. Fin /\ B = |^|y)) -> E.z(z (_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
26		sppfi 10472	. . . . . . . 8 |- ((A e. (fi` C) /\ C e. D) -> (A e. (fi` C) <-> E.x(x (_ C /\ x e. Fin /\ A = |^|x)))
27	26	biimpd 153	. . . . . . 7 |- ((A e. (fi` C) /\ C e. D) -> (A e. (fi` C) -> E.x(x (_ C /\ x e. Fin /\ A = |^|x)))
28	27	ex 373	. . . . . 6 |- (A e. (fi` C) -> (C e. D -> (A e. (fi` C) -> E.x(x (_ C /\ x e. Fin /\ A = |^|x))))
29	28	pm2.43b 67	. . . . 5 |- (C e. D -> (A e. (fi` C) -> E.x(x (_ C /\ x e. Fin /\ A = |^|x)))
30		sppfi 10472	. . . . . . . 8 |- ((B e. (fi` C) /\ C e. D) -> (B e. (fi` C) <-> E.y(y (_ C /\ y e. Fin /\ B = |^|y)))
31	30	biimpd 153	. . . . . . 7 |- ((B e. (fi` C) /\ C e. D) -> (B e. (fi` C) -> E.y(y (_ C /\ y e. Fin /\ B = |^|y)))
32	31	ex 373	. . . . . 6 |- (B e. (fi` C) -> (C e. D -> (B e. (fi` C) -> E.y(y (_ C /\ y e. Fin /\ B = |^|y))))
33	32	pm2.43b 67	. . . . 5 |- (C e. D -> (B e. (fi` C) -> E.y(y (_ C /\ y e. Fin /\ B = |^|y)))
34	25, 29, 33	syl2and 461	. . . 4 |- (C e. D -> ((A e. (fi` C) /\ B e. (fi` C)) -> E.z(z (_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
35	34	imp 350	. . 3 |- ((C e. D /\ (A e. (fi` C) /\ B e. (fi` C))) -> E.z(z (_ C /\ z e. Fin /\ (A i^i B) = |^|z))
36		inex1g 2723	. . . . . . 7 |- (A e. (fi` C) -> (A i^i B) e. V)
37	36	adantr 391	. . . . . 6 |- ((A e. (fi` C) /\ B e. (fi` C)) -> (A i^i B) e. V)
38	37	anim1i 334	. . . . 5 |- (((A e. (fi` C) /\ B e. (fi` C)) /\ C e. D) -> ((A i^i B) e. V /\ C e. D))
39	38	ancoms 438	. . . 4 |- ((C e. D /\ (A e. (fi` C) /\ B e. (fi` C))) -> ((A i^i B) e. V /\ C e. D))
40		sppfi 10472	. . . 4 |- (((A i^i B) e. V /\ C e. D) -> ((A i^i B) e. (fi` C) <-> E.z(z (_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
41	39, 40	syl 10	. . 3 |- ((C e. D /\ (A e. (fi` C) /\ B e. (fi` C))) -> ((A i^i B) e. (fi` C) <-> E.z(z (_ C /\ z e. Fin /\ (A i^i B) = |^|z)))
42	35, 41	mpbird 196	. 2 |- ((C e. D /\ (A e. (fi` C) /\ B e. (fi` C))) -> (A i^i B) e. (fi` C))
43	42	ex 373	1 |- (C e. D -> ((A e. (fi` C) /\ B e. (fi` C)) -> (A i^i B) e. (fi` C)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  E.wex 982  Vcvv 1814   u. cun 2048   i^i cin 2049   (_ wss 2050  |^|cint 2537  ` cfv 3188  Fincfn 4373  ficfi 10468

This theorem is referenced by:  fgsb2 10560

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-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  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-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-csb 2005  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-pss 2058  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  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-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-opr 3971  df-oprab 3972  df-oadd 4141  df-er 4267  df-en 4374  df-fin 4377  df-fi 10469

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