Theorem abfii2 4536

Description: Two ways to express the collection of finite intersections of a set A.

Hypothesis

Ref	Expression
abfii2.1	|- A e. V

Assertion

Ref	Expression
abfii2	|- {x | E.y(y (_ A /\ E.z e. om y ~~ z /\ x = |^|y)} = |^|{x | A.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> |^|y e. x)}

Distinct variable group:   x,y,z,A

Proof of Theorem abfii2

Step	Hyp	Ref	Expression
1		abfii2.1	. . . . 5 |- A e. V
2	1	uniex 2861	. . . 4 |- U.A e. V
3	2	inex2 2707	. . 3 |- (|^|y i^i U.A) e. V
4		df-sn 2402	. . . . . 6 |- {(|^|y i^i U.A)} = {x | x = (|^|y i^i U.A)}
5		snex 2740	. . . . . 6 |- {(|^|y i^i U.A)} e. V
6	4, 5	eqeltrr 1537	. . . . 5 |- {x | x = (|^|y i^i U.A)} e. V
7	1, 6	abexssex 3857	. . . 4 |- {x | E.y(y (_ A /\ x = (|^|y i^i U.A))} e. V
8		3simp1 786	. . . . . . 7 |- ((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> y (_ A)
9	8	anim1i 334	. . . . . 6 |- (((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) /\ x = (|^|y i^i U.A)) -> (y (_ A /\ x = (|^|y i^i U.A)))
10	9	19.22i 1036	. . . . 5 |- (E.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) /\ x = (|^|y i^i U.A)) -> E.y(y (_ A /\ x = (|^|y i^i U.A)))
11	10	ss2abi 2110	. . . 4 |- {x | E.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) /\ x = (|^|y i^i U.A))} (_ {x | E.y(y (_ A /\ x = (|^|y i^i U.A))}
12	7, 11	ssexi 2710	. . 3 |- {x | E.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) /\ x = (|^|y i^i U.A))} e. V
13	3, 12	intab 2550	. 2 |- |^|{x | A.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> (|^|y i^i U.A) e. x)} = {x | E.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) /\ x = (|^|y i^i U.A))}
14		intssuni2 2546	. . . . . . . . 9 |- ((y (_ A /\ y =/= (/)) -> |^|y (_ U.A)
15		dfss 2044	. . . . . . . . 9 |- (|^|y (_ U.A <-> |^|y = (|^|y i^i U.A))
16	14, 15	sylib 198	. . . . . . . 8 |- ((y (_ A /\ y =/= (/)) -> |^|y = (|^|y i^i U.A))
17	16	3adant3 797	. . . . . . 7 |- ((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> |^|y = (|^|y i^i U.A))
18	17	eleq1d 1532	. . . . . 6 |- ((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> (|^|y e. x <-> (|^|y i^i U.A) e. x))
19	18	pm5.74i 582	. . . . 5 |- (((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> |^|y e. x) <-> ((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> (|^|y i^i U.A) e. x))
20	19	albii 996	. . . 4 |- (A.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> |^|y e. x) <-> A.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> (|^|y i^i U.A) e. x))
21	20	abbii 1567	. . 3 |- {x | A.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> |^|y e. x)} = {x | A.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> (|^|y i^i U.A) e. x)}
22	21	inteqi 2527	. 2 |- |^|{x | A.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> |^|y e. x)} = |^|{x | A.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> (|^|y i^i U.A) e. x)}
23		df-3an 775	. . . . 5 |- ((y (_ A /\ E.z e. om y ~~ z /\ x = |^|y) <-> ((y (_ A /\ E.z e. om y ~~ z) /\ x = |^|y))
24		visset 1804	. . . . . . . . 9 |- x e. V
25		eleq1 1526	. . . . . . . . 9 |- (x = |^|y -> (x e. V <-> |^|y e. V))
26	24, 25	mpbii 193	. . . . . . . 8 |- (x = |^|y -> |^|y e. V)
27		intex 2719	. . . . . . . 8 |- (y =/= (/) <-> |^|y e. V)
28	26, 27	sylibr 200	. . . . . . 7 |- (x = |^|y -> y =/= (/))
29	28	pm4.71ri 636	. . . . . 6 |- (x = |^|y <-> (y =/= (/) /\ x = |^|y))
30	29	anbi2i 479	. . . . 5 |- (((y (_ A /\ E.z e. om y ~~ z) /\ x = |^|y) <-> ((y (_ A /\ E.z e. om y ~~ z) /\ (y =/= (/) /\ x = |^|y)))
31		an4 505	. . . . . 6 |- (((y (_ A /\ E.z e. om y ~~ z) /\ (y =/= (/) /\ x = |^|y)) <-> ((y (_ A /\ y =/= (/)) /\ (E.z e. om y ~~ z /\ x = |^|y)))
32		df-3an 775	. . . . . . . 8 |- ((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) <-> ((y (_ A /\ y =/= (/)) /\ E.z e. om y ~~ z))
33	32	anbi1i 480	. . . . . . 7 |- (((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) /\ x = |^|y) <-> (((y (_ A /\ y =/= (/)) /\ E.z e. om y ~~ z) /\ x = |^|y))
34	16	eqeq2d 1478	. . . . . . . . 9 |- ((y (_ A /\ y =/= (/)) -> (x = |^|y <-> x = (|^|y i^i U.A)))
35	34	3adant3 797	. . . . . . . 8 |- ((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> (x = |^|y <-> x = (|^|y i^i U.A)))
36	35	pm5.32i 643	. . . . . . 7 |- (((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) /\ x = |^|y) <-> ((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) /\ x = (|^|y i^i U.A)))
37		anass 439	. . . . . . 7 |- ((((y (_ A /\ y =/= (/)) /\ E.z e. om y ~~ z) /\ x = |^|y) <-> ((y (_ A /\ y =/= (/)) /\ (E.z e. om y ~~ z /\ x = |^|y)))
38	33, 36, 37	3bitr3r 182	. . . . . 6 |- (((y (_ A /\ y =/= (/)) /\ (E.z e. om y ~~ z /\ x = |^|y)) <-> ((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) /\ x = (|^|y i^i U.A)))
39	31, 38	bitr 173	. . . . 5 |- (((y (_ A /\ E.z e. om y ~~ z) /\ (y =/= (/) /\ x = |^|y)) <-> ((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) /\ x = (|^|y i^i U.A)))
40	23, 30, 39	3bitr 177	. . . 4 |- ((y (_ A /\ E.z e. om y ~~ z /\ x = |^|y) <-> ((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) /\ x = (|^|y i^i U.A)))
41	40	exbii 1047	. . 3 |- (E.y(y (_ A /\ E.z e. om y ~~ z /\ x = |^|y) <-> E.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) /\ x = (|^|y i^i U.A)))
42	41	abbii 1567	. 2 |- {x | E.y(y (_ A /\ E.z e. om y ~~ z /\ x = |^|y)} = {x | E.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) /\ x = (|^|y i^i U.A))}
43	13, 22, 42	3eqtr4r 1498	1 |- {x | E.y(y (_ A /\ E.z e. om y ~~ z /\ x = |^|y)} = |^|{x | A.y((y (_ A /\ y =/= (/) /\ E.z e. om y ~~ z) -> |^|y e. x)}

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773  A.wal 951   = wceq 953   e. wcel 955  E.wex 977  {cab 1456   =/= wne 1577  E.wrex 1638  Vcvv 1802   i^i cin 2036   (_ wss 2037  (/)c0 2270  {csn 2399  U.cuni 2493  |^|cint 2523   class class class wbr 2609  omcom 3121   ~~ cen 4348

This theorem is referenced by:  abfii3 4537  abfii4 4538  abfii5 4539

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-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-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-id 2824  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-fv 3188

Copyright terms: Public domain