Theorem fiiu 10453

Description: If A is the intersection of a finite set of elements of B then A (_ U.B.

Assertion

Ref	Expression
fiiu	|- (A e. {x | E.y(y (_ B /\ y e. Fin /\ x = |^|y)} -> A (_ U.B)

Distinct variable groups:   x,A,y   x,B,y

Proof of Theorem fiiu

Step	Hyp	Ref	Expression
1		nvel 2714	. . . 4 |- -. V e. {x | E.y(y (_ B /\ y e. Fin /\ x = |^|y)}
2		eleq1 1534	. . . . 5 |- (A = V -> (A e. {x | E.y(y (_ B /\ y e. Fin /\ x = |^|y)} <-> V e. {x | E.y(y (_ B /\ y e. Fin /\ x = |^|y)}))
3	2	biimpcd 155	. . . 4 |- (A e. {x | E.y(y (_ B /\ y e. Fin /\ x = |^|y)} -> (A = V -> V e. {x | E.y(y (_ B /\ y e. Fin /\ x = |^|y)}))
4	1, 3	mtoi 107	. . 3 |- (A e. {x | E.y(y (_ B /\ y e. Fin /\ x = |^|y)} -> -. A = V)
5		spfi 10445	. . . 4 |- (A e. {x | E.y(y (_ B /\ y e. Fin /\ x = |^|y)} -> (A e. {x | E.y(y (_ B /\ y e. Fin /\ x = |^|y)} <-> E.y(y (_ B /\ y e. Fin /\ A = |^|y)))
6		eqeq1 1481	. . . . . . . . . . . 12 |- (A = |^|y -> (A = V <-> |^|y = V))
7	6	negbid 611	. . . . . . . . . . 11 |- (A = |^|y -> (-. A = V <-> -. |^|y = V))
8	7	biimpd 153	. . . . . . . . . 10 |- (A = |^|y -> (-. A = V -> -. |^|y = V))
9		int0 2547	. . . . . . . . . . 11 |- |^|(/) = V
10		eqeq2 1484	. . . . . . . . . . . . . 14 |- (|^|(/) = V -> (|^|y = |^|(/) <-> |^|y = V))
11	10	bicomd 521	. . . . . . . . . . . . 13 |- (|^|(/) = V -> (|^|y = V <-> |^|y = |^|(/)))
12	11	negbid 611	. . . . . . . . . . . 12 |- (|^|(/) = V -> (-. |^|y = V <-> -. |^|y = |^|(/)))
13		inteq 2536	. . . . . . . . . . . . . 14 |- (y = (/) -> |^|y = |^|(/))
14	13	con3i 98	. . . . . . . . . . . . 13 |- (-. |^|y = |^|(/) -> -. y = (/))
15		sseq1 2082	. . . . . . . . . . . . . . . . . 18 |- (|^|y = A -> (|^|y (_ U.B <-> A (_ U.B))
16	15	biimpd 153	. . . . . . . . . . . . . . . . 17 |- (|^|y = A -> (|^|y (_ U.B -> A (_ U.B))
17	16	eqcoms 1478	. . . . . . . . . . . . . . . 16 |- (A = |^|y -> (|^|y (_ U.B -> A (_ U.B))
18		intssuni2 2556	. . . . . . . . . . . . . . . . . . 19 |- ((y (_ B /\ y =/= (/)) -> |^|y (_ U.B)
19		df-ne 1587	. . . . . . . . . . . . . . . . . . 19 |- (y =/= (/) <-> -. y = (/))
20	18, 19	sylan2br 453	. . . . . . . . . . . . . . . . . 18 |- ((y (_ B /\ -. y = (/)) -> |^|y (_ U.B)
21	20	ancoms 436	. . . . . . . . . . . . . . . . 17 |- ((-. y = (/) /\ y (_ B) -> |^|y (_ U.B)
22	21	3adant2 798	. . . . . . . . . . . . . . . 16 |- ((-. y = (/) /\ y e. Fin /\ y (_ B) -> |^|y (_ U.B)
23	17, 22	syl5com 52	. . . . . . . . . . . . . . 15 |- ((-. y = (/) /\ y e. Fin /\ y (_ B) -> (A = |^|y -> A (_ U.B))
24	23	3exp 832	. . . . . . . . . . . . . 14 |- (-. y = (/) -> (y e. Fin -> (y (_ B -> (A = |^|y -> A (_ U.B))))
25	24	com24 37	. . . . . . . . . . . . 13 |- (-. y = (/) -> (A = |^|y -> (y (_ B -> (y e. Fin -> A (_ U.B))))
26	14, 25	syl 10	. . . . . . . . . . . 12 |- (-. |^|y = |^|(/) -> (A = |^|y -> (y (_ B -> (y e. Fin -> A (_ U.B))))
27	12, 26	syl6bi 214	. . . . . . . . . . 11 |- (|^|(/) = V -> (-. |^|y = V -> (A = |^|y -> (y (_ B -> (y e. Fin -> A (_ U.B)))))
28	9, 27	ax-mp 7	. . . . . . . . . 10 |- (-. |^|y = V -> (A = |^|y -> (y (_ B -> (y e. Fin -> A (_ U.B))))
29	8, 28	syl6com 53	. . . . . . . . 9 |- (-. A = V -> (A = |^|y -> (A = |^|y -> (y (_ B -> (y e. Fin -> A (_ U.B)))))
30	29	com13 33	. . . . . . . 8 |- (A = |^|y -> (A = |^|y -> (-. A = V -> (y (_ B -> (y e. Fin -> A (_ U.B)))))
31	30	pm2.43i 64	. . . . . . 7 |- (A = |^|y -> (-. A = V -> (y (_ B -> (y e. Fin -> A (_ U.B))))
32	31	com4t 40	. . . . . 6 |- (y (_ B -> (y e. Fin -> (A = |^|y -> (-. A = V -> A (_ U.B))))
33	32	3imp 827	. . . . 5 |- ((y (_ B /\ y e. Fin /\ A = |^|y) -> (-. A = V -> A (_ U.B))
34	33	19.23aiv 1295	. . . 4 |- (E.y(y (_ B /\ y e. Fin /\ A = |^|y) -> (-. A = V -> A (_ U.B))
35	5, 34	syl6bi 214	. . 3 |- (A e. {x | E.y(y (_ B /\ y e. Fin /\ x = |^|y)} -> (A e. {x | E.y(y (_ B /\ y e. Fin /\ x = |^|y)} -> (-. A = V -> A (_ U.B)))
36	4, 35	mpid 47	. 2 |- (A e. {x | E.y(y (_ B /\ y e. Fin /\ x = |^|y)} -> (A e. {x | E.y(y (_ B /\ y e. Fin /\ x = |^|y)} -> A (_ U.B))
37	36	pm2.43i 64	1 |- (A e. {x | E.y(y (_ B /\ y e. Fin /\ x = |^|y)} -> A (_ U.B)

Syntax hints:  -. wn 2   -> wi 3   /\ w3a 775   = wceq 956   e. wcel 958  E.wex 980  {cab 1463   =/= wne 1585  Vcvv 1811   (_ wss 2047  (/)c0 2280  U.cuni 2503  |^|cint 2533  Fincfn 4367

This theorem is referenced by:  fgsb 10570

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-10 966  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

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281  df-uni 2504  df-int 2534

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