Theorem fiiu2 10488

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

Assertion

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

Proof of Theorem fiiu2

Step	Hyp	Ref	Expression
1		nvel 2714	. . . . 5 |- -. V e. (fi` B)
2		eleq1 1534	. . . . . 6 |- (A = V -> (A e. (fi` B) <-> V e. (fi` B)))
3	2	biimpcd 155	. . . . 5 |- (A e. (fi` B) -> (A = V -> V e. (fi` B)))
4	1, 3	mtoi 107	. . . 4 |- (A e. (fi` B) -> -. A = V)
5		sppfi 10486	. . . . . . . 8 |- ((A e. (fi` B) /\ B e. C) -> (A e. (fi` B) <-> E.y(y (_ B /\ y e. Fin /\ A = |^|y)))
6		eqeq1 1481	. . . . . . . . . . . . . . . 16 |- (A = |^|y -> (A = V <-> |^|y = V))
7	6	negbid 611	. . . . . . . . . . . . . . 15 |- (A = |^|y -> (-. A = V <-> -. |^|y = V))
8		int0 2547	. . . . . . . . . . . . . . . 16 |- |^|(/) = V
9		neeq2 1591	. . . . . . . . . . . . . . . . . 18 |- (|^|(/) = V -> (|^|y =/= |^|(/) <-> |^|y =/= V))
10		inteq 2536	. . . . . . . . . . . . . . . . . . 19 |- (y = (/) -> |^|y = |^|(/))
11	10	necon3i 1605	. . . . . . . . . . . . . . . . . 18 |- (|^|y =/= |^|(/) -> y =/= (/))
12	9, 11	syl6bir 215	. . . . . . . . . . . . . . . . 17 |- (|^|(/) = V -> (|^|y =/= V -> y =/= (/)))
13		df-ne 1587	. . . . . . . . . . . . . . . . 17 |- (|^|y =/= V <-> -. |^|y = V)
14	12, 13	syl5ibr 207	. . . . . . . . . . . . . . . 16 |- (|^|(/) = V -> (-. |^|y = V -> y =/= (/)))
15	8, 14	ax-mp 7	. . . . . . . . . . . . . . 15 |- (-. |^|y = V -> y =/= (/))
16	7, 15	syl6bi 214	. . . . . . . . . . . . . 14 |- (A = |^|y -> (-. A = V -> y =/= (/)))
17		sseq1 2082	. . . . . . . . . . . . . . . . . 18 |- (|^|y = A -> (|^|y (_ U.B <-> A (_ U.B))
18	17	biimpd 153	. . . . . . . . . . . . . . . . 17 |- (|^|y = A -> (|^|y (_ U.B -> A (_ U.B))
19	18	eqcoms 1478	. . . . . . . . . . . . . . . 16 |- (A = |^|y -> (|^|y (_ U.B -> A (_ U.B))
20		intssuni2 2556	. . . . . . . . . . . . . . . 16 |- ((y (_ B /\ y =/= (/)) -> |^|y (_ U.B)
21	19, 20	syl5com 52	. . . . . . . . . . . . . . 15 |- ((y (_ B /\ y =/= (/)) -> (A = |^|y -> A (_ U.B))
22	21	expcom 374	. . . . . . . . . . . . . 14 |- (y =/= (/) -> (y (_ B -> (A = |^|y -> A (_ U.B)))
23	16, 22	syl6 22	. . . . . . . . . . . . 13 |- (A = |^|y -> (-. A = V -> (y (_ B -> (A = |^|y -> A (_ U.B))))
24	23	com24 37	. . . . . . . . . . . 12 |- (A = |^|y -> (A = |^|y -> (y (_ B -> (-. A = V -> A (_ U.B))))
25	24	pm2.43i 64	. . . . . . . . . . 11 |- (A = |^|y -> (y (_ B -> (-. A = V -> A (_ U.B)))
26	25	impcom 351	. . . . . . . . . 10 |- ((y (_ B /\ A = |^|y) -> (-. A = V -> A (_ U.B))
27	26	3adant2 798	. . . . . . . . 9 |- ((y (_ B /\ y e. Fin /\ A = |^|y) -> (-. A = V -> A (_ U.B))
28	27	19.23aiv 1295	. . . . . . . 8 |- (E.y(y (_ B /\ y e. Fin /\ A = |^|y) -> (-. A = V -> A (_ U.B))
29	5, 28	syl6bi 214	. . . . . . 7 |- ((A e. (fi` B) /\ B e. C) -> (A e. (fi` B) -> (-. A = V -> A (_ U.B)))
30	29	ex 373	. . . . . 6 |- (A e. (fi` B) -> (B e. C -> (A e. (fi` B) -> (-. A = V -> A (_ U.B))))
31	30	com23 32	. . . . 5 |- (A e. (fi` B) -> (A e. (fi` B) -> (B e. C -> (-. A = V -> A (_ U.B))))
32	31	com34 36	. . . 4 |- (A e. (fi` B) -> (A e. (fi` B) -> (-. A = V -> (B e. C -> A (_ U.B))))
33	4, 32	mpid 47	. . 3 |- (A e. (fi` B) -> (A e. (fi` B) -> (B e. C -> A (_ U.B)))
34	33	pm2.43i 64	. 2 |- (A e. (fi` B) -> (B e. C -> A (_ U.B))
35	34	com12 11	1 |- (B e. C -> (A e. (fi` B) -> A (_ U.B))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wex 980   =/= wne 1585  Vcvv 1811   (_ wss 2047  (/)c0 2280  U.cuni 2503  |^|cint 2533  ` cfv 3182  Fincfn 4367  ficfi 10479

This theorem is referenced by:  fgsb2 10580

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-9 965  ax-10 966  ax-11 967  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  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-int 2534  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-fi 10480

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