Theorem infi 10448

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

Assertion

Ref	Expression
infi	⊢ (C ∈ D → ((A ∈ (fi ‘C) ⋀ B ∈ (fi ‘C)) → (A ∩ B) ∈ (fi ‘C)))

Proof of Theorem infi

Step	Hyp	Ref	Expression
1		an6 899	. . . . . . . . . . . . 13 ⊢ (((x ⊆ C ⋀ ∃a ∈ ω x ≈ a ⋀ A = ∩x) ⋀ (y ⊆ C ⋀ ∃b ∈ ω y ≈ b ⋀ B = ∩y)) ↔ ((x ⊆ C ⋀ y ⊆ C) ⋀ (∃a ∈ ω x ≈ a ⋀ ∃b ∈ ω y ≈ b) ⋀ (A = ∩x ⋀ B = ∩y)))
2		unss 2194	. . . . . . . . . . . . . . 15 ⊢ ((x ⊆ C ⋀ y ⊆ C) ↔ (x ∪ y) ⊆ C)
3	2	biimp 151	. . . . . . . . . . . . . 14 ⊢ ((x ⊆ C ⋀ y ⊆ C) → (x ∪ y) ⊆ C)
4		unfi 4528	. . . . . . . . . . . . . . 15 ⊢ ((∃a ∈ ω x ≈ a ⋀ ∃a ∈ ω y ≈ a) → ∃a ∈ ω (x ∪ y) ≈ a)
5		breq2 2613	. . . . . . . . . . . . . . . . 17 ⊢ (b = a → (y ≈ b ↔ y ≈ a))
6	5	cbvrexv 1792	. . . . . . . . . . . . . . . 16 ⊢ (∃b ∈ ω y ≈ b ↔ ∃a ∈ ω y ≈ a)
7	6	anbi2i 479	. . . . . . . . . . . . . . 15 ⊢ ((∃a ∈ ω x ≈ a ⋀ ∃b ∈ ω y ≈ b) ↔ (∃a ∈ ω x ≈ a ⋀ ∃a ∈ ω y ≈ a))
8		breq2 2613	. . . . . . . . . . . . . . . 16 ⊢ (c = a → ((x ∪ y) ≈ c ↔ (x ∪ y) ≈ a))
9	8	cbvrexv 1792	. . . . . . . . . . . . . . 15 ⊢ (∃c ∈ ω (x ∪ y) ≈ c ↔ ∃a ∈ ω (x ∪ y) ≈ a)
10	4, 7, 9	3imtr4 219	. . . . . . . . . . . . . 14 ⊢ ((∃a ∈ ω x ≈ a ⋀ ∃b ∈ ω y ≈ b) → ∃c ∈ ω (x ∪ y) ≈ c)
11		ineq12 2202	. . . . . . . . . . . . . . 15 ⊢ ((A = ∩x ⋀ B = ∩y) → (A ∩ B) = (∩x ∩ ∩y))
12		intun 2552	. . . . . . . . . . . . . . 15 ⊢ ∩(x ∪ y) = (∩x ∩ ∩y)
13	11, 12	syl6eqr 1517	. . . . . . . . . . . . . 14 ⊢ ((A = ∩x ⋀ B = ∩y) → (A ∩ B) = ∩(x ∪ y))
14	3, 10, 13	3anim123i 819	. . . . . . . . . . . . 13 ⊢ (((x ⊆ C ⋀ y ⊆ C) ⋀ (∃a ∈ ω x ≈ a ⋀ ∃b ∈ ω y ≈ b) ⋀ (A = ∩x ⋀ B = ∩y)) → ((x ∪ y) ⊆ C ⋀ ∃c ∈ ω (x ∪ y) ≈ c ⋀ (A ∩ B) = ∩(x ∪ y)))
15	1, 14	sylbi 199	. . . . . . . . . . . 12 ⊢ (((x ⊆ C ⋀ ∃a ∈ ω x ≈ a ⋀ A = ∩x) ⋀ (y ⊆ C ⋀ ∃b ∈ ω y ≈ b ⋀ B = ∩y)) → ((x ∪ y) ⊆ C ⋀ ∃c ∈ ω (x ∪ y) ≈ c ⋀ (A ∩ B) = ∩(x ∪ y)))
16	15	ex 373	. . . . . . . . . . 11 ⊢ ((x ⊆ C ⋀ ∃a ∈ ω x ≈ a ⋀ A = ∩x) → ((y ⊆ C ⋀ ∃b ∈ ω y ≈ b ⋀ B = ∩y) → ((x ∪ y) ⊆ C ⋀ ∃c ∈ ω (x ∪ y) ≈ c ⋀ (A ∩ B) = ∩(x ∪ y))))
17		visset 1804	. . . . . . . . . . . . 13 ⊢ x ∈ V
18		visset 1804	. . . . . . . . . . . . 13 ⊢ y ∈ V
19	17, 18	unex 2863	. . . . . . . . . . . 12 ⊢ (x ∪ y) ∈ V
20		sseq1 2072	. . . . . . . . . . . . 13 ⊢ (z = (x ∪ y) → (z ⊆ C ↔ (x ∪ y) ⊆ C))
21		breq1 2612	. . . . . . . . . . . . . 14 ⊢ (z = (x ∪ y) → (z ≈ c ↔ (x ∪ y) ≈ c))
22	21	rexbidv 1656	. . . . . . . . . . . . 13 ⊢ (z = (x ∪ y) → (∃c ∈ ω z ≈ c ↔ ∃c ∈ ω (x ∪ y) ≈ c))
23		inteq 2526	. . . . . . . . . . . . . 14 ⊢ (z = (x ∪ y) → ∩z = ∩(x ∪ y))
24	23	eqeq2d 1478	. . . . . . . . . . . . 13 ⊢ (z = (x ∪ y) → ((A ∩ B) = ∩z ↔ (A ∩ B) = ∩(x ∪ y)))
25	20, 22, 24	3anbi123d 890	. . . . . . . . . . . 12 ⊢ (z = (x ∪ y) → ((z ⊆ C ⋀ ∃c ∈ ω z ≈ c ⋀ (A ∩ B) = ∩z) ↔ ((x ∪ y) ⊆ C ⋀ ∃c ∈ ω (x ∪ y) ≈ c ⋀ (A ∩ B) = ∩(x ∪ y))))
26	19, 25	cla4ev 1860	. . . . . . . . . . 11 ⊢ (((x ∪ y) ⊆ C ⋀ ∃c ∈ ω (x ∪ y) ≈ c ⋀ (A ∩ B) = ∩(x ∪ y)) → ∃z(z ⊆ C ⋀ ∃c ∈ ω z ≈ c ⋀ (A ∩ B) = ∩z))
27	16, 26	syl6com 53	. . . . . . . . . 10 ⊢ ((y ⊆ C ⋀ ∃b ∈ ω y ≈ b ⋀ B = ∩y) → ((x ⊆ C ⋀ ∃a ∈ ω x ≈ a ⋀ A = ∩x) → ∃z(z ⊆ C ⋀ ∃c ∈ ω z ≈ c ⋀ (A ∩ B) = ∩z)))
28	27	19.23aiv 1290	. . . . . . . . 9 ⊢ (∃y(y ⊆ C ⋀ ∃b ∈ ω y ≈ b ⋀ B = ∩y) → ((x ⊆ C ⋀ ∃a ∈ ω x ≈ a ⋀ A = ∩x) → ∃z(z ⊆ C ⋀ ∃c ∈ ω z ≈ c ⋀ (A ∩ B) = ∩z)))
29	28	com12 11	. . . . . . . 8 ⊢ ((x ⊆ C ⋀ ∃a ∈ ω x ≈ a ⋀ A = ∩x) → (∃y(y ⊆ C ⋀ ∃b ∈ ω y ≈ b ⋀ B = ∩y) → ∃z(z ⊆ C ⋀ ∃c ∈ ω z ≈ c ⋀ (A ∩ B) = ∩z)))
30	29	19.23aiv 1290	. . . . . . 7 ⊢ (∃x(x ⊆ C ⋀ ∃a ∈ ω x ≈ a ⋀ A = ∩x) → (∃y(y ⊆ C ⋀ ∃b ∈ ω y ≈ b ⋀ B = ∩y) → ∃z(z ⊆ C ⋀ ∃c ∈ ω z ≈ c ⋀ (A ∩ B) = ∩z)))
31	30	imp 350	. . . . . 6 ⊢ ((∃x(x ⊆ C ⋀ ∃a ∈ ω x ≈ a ⋀ A = ∩x) ⋀ ∃y(y ⊆ C ⋀ ∃b ∈ ω y ≈ b ⋀ B = ∩y)) → ∃z(z ⊆ C ⋀ ∃c ∈ ω z ≈ c ⋀ (A ∩ B) = ∩z))
32	31	a1i 8	. . . . 5 ⊢ (C ∈ D → ((∃x(x ⊆ C ⋀ ∃a ∈ ω x ≈ a ⋀ A = ∩x) ⋀ ∃y(y ⊆ C ⋀ ∃b ∈ ω y ≈ b ⋀ B = ∩y)) → ∃z(z ⊆ C ⋀ ∃c ∈ ω z ≈ c ⋀ (A ∩ B) = ∩z)))
33		sppfi 10376	. . . . . . . 8 ⊢ ((A ∈ (fi ‘C) ⋀ C ∈ D) → (A ∈ (fi ‘C) ↔ ∃x(x ⊆ C ⋀ ∃a ∈ ω x ≈ a ⋀ A = ∩x)))
34	33	biimpd 153	. . . . . . 7 ⊢ ((A ∈ (fi ‘C) ⋀ C ∈ D) → (A ∈ (fi ‘C) → ∃x(x ⊆ C ⋀ ∃a ∈ ω x ≈ a ⋀ A = ∩x)))
35	34	ex 373	. . . . . 6 ⊢ (A ∈ (fi ‘C) → (C ∈ D → (A ∈ (fi ‘C) → ∃x(x ⊆ C ⋀ ∃a ∈ ω x ≈ a ⋀ A = ∩x))))
36	35	pm2.43b 67	. . . . 5 ⊢ (C ∈ D → (A ∈ (fi ‘C) → ∃x(x ⊆ C ⋀ ∃a ∈ ω x ≈ a ⋀ A = ∩x)))
37		sppfi 10376	. . . . . . . 8 ⊢ ((B ∈ (fi ‘C) ⋀ C ∈ D) → (B ∈ (fi ‘C) ↔ ∃y(y ⊆ C ⋀ ∃b ∈ ω y ≈ b ⋀ B = ∩y)))
38	37	biimpd 153	. . . . . . 7 ⊢ ((B ∈ (fi ‘C) ⋀ C ∈ D) → (B ∈ (fi ‘C) → ∃y(y ⊆ C ⋀ ∃b ∈ ω y ≈ b ⋀ B = ∩y)))
39	38	ex 373	. . . . . 6 ⊢ (B ∈ (fi ‘C) → (C ∈ D → (B ∈ (fi ‘C) → ∃y(y ⊆ C ⋀ ∃b ∈ ω y ≈ b ⋀ B = ∩y))))
40	39	pm2.43b 67	. . . . 5 ⊢ (C ∈ D → (B ∈ (fi ‘C) → ∃y(y ⊆ C ⋀ ∃b ∈ ω y ≈ b ⋀ B = ∩y)))
41	32, 36, 40	syl2and 459	. . . 4 ⊢ (C ∈ D → ((A ∈ (fi ‘C) ⋀ B ∈ (fi ‘C)) → ∃z(z ⊆ C ⋀ ∃c ∈ ω z ≈ c ⋀ (A ∩ B) = ∩z)))
42	41	imp 350	. . 3 ⊢ ((C ∈ D ⋀ (A ∈ (fi ‘C) ⋀ B ∈ (fi ‘C))) → ∃z(z ⊆ C ⋀ ∃c ∈ ω z ≈ c ⋀ (A ∩ B) = ∩z))
43		inex1g 2708	. . . . . . 7 ⊢ (A ∈ (fi ‘C) → (A ∩ B) ∈ V)
44	43	adantr 389	. . . . . 6 ⊢ ((A ∈ (fi ‘C) ⋀ B ∈ (fi ‘C)) → (A ∩ B) ∈ V)
45	44	anim1i 334	. . . . 5 ⊢ (((A ∈ (fi ‘C) ⋀ B ∈ (fi ‘C)) ⋀ C ∈ D) → ((A ∩ B) ∈ V ⋀ C ∈ D))
46	45	ancoms 436	. . . 4 ⊢ ((C ∈ D ⋀ (A ∈ (fi ‘C) ⋀ B ∈ (fi ‘C))) → ((A ∩ B) ∈ V ⋀ C ∈ D))
47		sppfi 10376	. . . 4 ⊢ (((A ∩ B) ∈ V ⋀ C ∈ D) → ((A ∩ B) ∈ (fi ‘C) ↔ ∃z(z ⊆ C ⋀ ∃c ∈ ω z ≈ c ⋀ (A ∩ B) = ∩z)))
48	46, 47	syl 10	. . 3 ⊢ ((C ∈ D ⋀ (A ∈ (fi ‘C) ⋀ B ∈ (fi ‘C))) → ((A ∩ B) ∈ (fi ‘C) ↔ ∃z(z ⊆ C ⋀ ∃c ∈ ω z ≈ c ⋀ (A ∩ B) = ∩z)))
49	42, 48	mpbird 196	. 2 ⊢ ((C ∈ D ⋀ (A ∈ (fi ‘C) ⋀ B ∈ (fi ‘C))) → (A ∩ B) ∈ (fi ‘C))
50	49	ex 373	1 ⊢ (C ∈ D → ((A ∈ (fi ‘C) ⋀ B ∈ (fi ‘C)) → (A ∩ B) ∈ (fi ‘C)))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   ⋀ w3a 773   = wceq 953   ∈ wcel 955  ∃wex 977  ∃wrex 1638  Vcvv 1802   ∪ cun 2035   ∩ cin 2036   ⊆ wss 2037  ∩cint 2523   class class class wbr 2609  ωcom 3121   ‘cfv 3172   ≈ cen 4348  ficfi 10372

This theorem is referenced by:  fgsb2 10449

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-oadd 4119  df-er 4245  df-en 4351  df-fiNEW 10373

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