Theorem fiiu2 10220

Description: If A is the intersection of a finite set of elements of B then A C_ U.B. (Contributed by FL, 2-Sep-2008.)

Assertion

Ref	Expression
fiiu2	|- (B e. C -> (A e. ( fi ` B) -> A C_ U.B))

Proof of Theorem fiiu2

Step	Hyp	Ref	Expression
1		nvel 3450	. . . . 5 |- -. _V e. ( fi ` B)
2		eleq1 1957	. . . . . 6 |- (A = _V -> (A e. ( fi ` B) <-> _V e. ( fi ` B)))
3	2	biimpcd 172	. . . . 5 |- (A e. ( fi ` B) -> (A = _V -> _V e. ( fi ` B)))
4	1, 3	mtoi 122	. . . 4 |- (A e. ( fi ` B) -> -. A = _V)
5		sppfi 10218	. . . . . . . 8 |- ((A e. ( fi ` B) /\ B e. C) -> (A e. ( fi ` B) <-> E.y(y C_ B /\ y e. Fin /\ A = |^|y)))
6		eqeq1 1890	. . . . . . . . . . . . . . . 16 |- (A = |^|y -> (A = _V <-> |^|y = _V))
7	6	notbid 673	. . . . . . . . . . . . . . 15 |- (A = |^|y -> (-. A = _V <-> -. |^|y = _V))
8		int0 3230	. . . . . . . . . . . . . . . 16 |- |^|(/) = _V
9		neeq2 2025	. . . . . . . . . . . . . . . . . 18 |- (|^|(/) = _V -> (|^|y =/= |^|(/) <-> |^|y =/= _V))
10		inteq 3217	. . . . . . . . . . . . . . . . . . 19 |- (y = (/) -> |^|y = |^|(/))
11	10	necon3i 2042	. . . . . . . . . . . . . . . . . 18 |- (|^|y =/= |^|(/) -> y =/= (/))
12	9, 11	syl6bir 232	. . . . . . . . . . . . . . . . 17 |- (|^|(/) = _V -> (|^|y =/= _V -> y =/= (/)))
13		df-ne 2019	. . . . . . . . . . . . . . . . 17 |- (|^|y =/= _V <-> -. |^|y = _V)
14	12, 13	syl5ibr 224	. . . . . . . . . . . . . . . 16 |- (|^|(/) = _V -> (-. |^|y = _V -> y =/= (/)))
15	8, 14	ax-mp 7	. . . . . . . . . . . . . . 15 |- (-. |^|y = _V -> y =/= (/))
16	7, 15	syl6bi 231	. . . . . . . . . . . . . 14 |- (A = |^|y -> (-. A = _V -> y =/= (/)))
17		sseq1 2637	. . . . . . . . . . . . . . . . . 18 |- (|^|y = A -> (|^|y C_ U.B <-> A C_ U.B))
18	17	biimpd 170	. . . . . . . . . . . . . . . . 17 |- (|^|y = A -> (|^|y C_ U.B -> A C_ U.B))
19	18	eqcoms 1887	. . . . . . . . . . . . . . . 16 |- (A = |^|y -> (|^|y C_ U.B -> A C_ U.B))
20		intssuni2 3243	. . . . . . . . . . . . . . . 16 |- ((y C_ B /\ y =/= (/)) -> |^|y C_ U.B)
21	19, 20	syl5com 63	. . . . . . . . . . . . . . 15 |- ((y C_ B /\ y =/= (/)) -> (A = |^|y -> A C_ U.B))
22	21	expcom 403	. . . . . . . . . . . . . 14 |- (y =/= (/) -> (y C_ B -> (A = |^|y -> A C_ U.B)))
23	16, 22	syl6 25	. . . . . . . . . . . . 13 |- (A = |^|y -> (-. A = _V -> (y C_ B -> (A = |^|y -> A C_ U.B))))
24	23	com24 41	. . . . . . . . . . . 12 |- (A = |^|y -> (A = |^|y -> (y C_ B -> (-. A = _V -> A C_ U.B))))
25	24	pm2.43i 78	. . . . . . . . . . 11 |- (A = |^|y -> (y C_ B -> (-. A = _V -> A C_ U.B)))
26	25	impcom 378	. . . . . . . . . 10 |- ((y C_ B /\ A = |^|y) -> (-. A = _V -> A C_ U.B))
27	26	3adant2 895	. . . . . . . . 9 |- ((y C_ B /\ y e. Fin /\ A = |^|y) -> (-. A = _V -> A C_ U.B))
28	27	19.23aiv 1674	. . . . . . . 8 |- (E.y(y C_ B /\ y e. Fin /\ A = |^|y) -> (-. A = _V -> A C_ U.B))
29	5, 28	syl6bi 231	. . . . . . 7 |- ((A e. ( fi ` B) /\ B e. C) -> (A e. ( fi ` B) -> (-. A = _V -> A C_ U.B)))
30	29	ex 402	. . . . . 6 |- (A e. ( fi ` B) -> (B e. C -> (A e. ( fi ` B) -> (-. A = _V -> A C_ U.B))))
31	30	com23 36	. . . . 5 |- (A e. ( fi ` B) -> (A e. ( fi ` B) -> (B e. C -> (-. A = _V -> A C_ U.B))))
32	31	com34 40	. . . 4 |- (A e. ( fi ` B) -> (A e. ( fi ` B) -> (-. A = _V -> (B e. C -> A C_ U.B))))
33	4, 32	mpid 58	. . 3 |- (A e. ( fi ` B) -> (A e. ( fi ` B) -> (B e. C -> A C_ U.B)))
34	33	pm2.43i 78	. 2 |- (A e. ( fi ` B) -> (B e. C -> A C_ U.B))
35	34	com12 14	1 |- (B e. C -> (A e. ( fi ` B) -> A C_ U.B))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  _Vcvv 2292   C_ wss 2593  (/)c0 2875  U.cuni 3177  |^|cint 3214  ` cfv 3998  Fincfn 5426   fi cfi 10210

This theorem is referenced by:  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-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-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-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-int 3215  df-br 3339  df-opab 3396  df-id 3586  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-fv 4014  df-fi 10211

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