Theorem abfi 10443

Description: Any element of A is the intersection of a finite subclass of A.

Assertion

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

Distinct variable group:   x,A,y

Proof of Theorem abfi

Step	Hyp	Ref	Expression
1		ssab 2121	. 2 |- (A (_ {x | E.y(y (_ A /\ y e. Fin /\ x = |^|y)} <-> A.x(x e. A -> E.y(y (_ A /\ y e. Fin /\ x = |^|y)))
2		snfi 4438	. . . 4 |- {x} e. Fin
3		visset 1816	. . . . . 6 |- x e. V
4	3	intsn 2568	. . . . 5 |- |^|{x} = x
5	4	eqcomi 1482	. . . 4 |- x = |^|{x}
6		eleq1 1537	. . . . . . 7 |- (y = {x} -> (y e. Fin <-> {x} e. Fin))
7		inteq 2540	. . . . . . . 8 |- (y = {x} -> |^|y = |^|{x})
8	7	eqeq2d 1489	. . . . . . 7 |- (y = {x} -> (x = |^|y <-> x = |^|{x}))
9	6, 8	anbi12d 630	. . . . . 6 |- (y = {x} -> ((y e. Fin /\ x = |^|y) <-> ({x} e. Fin /\ x = |^|{x})))
10	9	rcla4ev 1880	. . . . 5 |- (({x} e. P~A /\ ({x} e. Fin /\ x = |^|{x})) -> E.y e. P~ A(y e. Fin /\ x = |^|y))
11	10	ex 373	. . . 4 |- ({x} e. P~A -> (({x} e. Fin /\ x = |^|{x}) -> E.y e. P~ A(y e. Fin /\ x = |^|y)))
12	2, 5, 11	mp2ani 702	. . 3 |- ({x} e. P~A -> E.y e. P~ A(y e. Fin /\ x = |^|y))
13	3	snelpw 2758	. . 3 |- (x e. A <-> {x} e. P~A)
14		df-rex 1653	. . . 4 |- (E.y e. P~ A(y e. Fin /\ x = |^|y) <-> E.y(y e. P~A /\ (y e. Fin /\ x = |^|y)))
15		visset 1816	. . . . . . . 8 |- y e. V
16	15	elpw 2408	. . . . . . 7 |- (y e. P~A <-> y (_ A)
17	16	anbi1i 483	. . . . . 6 |- ((y e. P~A /\ (y e. Fin /\ x = |^|y)) <-> (y (_ A /\ (y e. Fin /\ x = |^|y)))
18		3anass 781	. . . . . 6 |- ((y (_ A /\ y e. Fin /\ x = |^|y) <-> (y (_ A /\ (y e. Fin /\ x = |^|y)))
19	17, 18	bitr4 176	. . . . 5 |- ((y e. P~A /\ (y e. Fin /\ x = |^|y)) <-> (y (_ A /\ y e. Fin /\ x = |^|y))
20	19	exbii 1053	. . . 4 |- (E.y(y e. P~A /\ (y e. Fin /\ x = |^|y)) <-> E.y(y (_ A /\ y e. Fin /\ x = |^|y))
21	14, 20	bitr2 174	. . 3 |- (E.y(y (_ A /\ y e. Fin /\ x = |^|y) <-> E.y e. P~ A(y e. Fin /\ x = |^|y))
22	12, 13, 21	3imtr4 219	. 2 |- (x e. A -> E.y(y (_ A /\ y e. Fin /\ x = |^|y))
23	1, 22	mpgbir 990	1 |- A (_ {x | E.y(y (_ A /\ y e. Fin /\ x = |^|y)}

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 777   = wceq 958   e. wcel 960  E.wex 982  {cab 1466  E.wrex 1649   (_ wss 2050  P~cpw 2405  {csn 2413  |^|cint 2537  Fincfn 4373

This theorem is referenced by:  abfi2 10474  efilcp 10556

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-1o 4139  df-en 4374  df-fin 4377

Copyright terms: Public domain