Theorem abfi 10451

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 2118	. 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 4432	. . . 4 |- {x} e. Fin
3		visset 1813	. . . . . 6 |- x e. V
4	3	intsn 2564	. . . . 5 |- |^|{x} = x
5	4	eqcomi 1479	. . . 4 |- x = |^|{x}
6		eleq1 1534	. . . . . . 7 |- (y = {x} -> (y e. Fin <-> {x} e. Fin))
7		inteq 2536	. . . . . . . 8 |- (y = {x} -> |^|y = |^|{x})
8	7	eqeq2d 1486	. . . . . . 7 |- (y = {x} -> (x = |^|y <-> x = |^|{x}))
9	6, 8	anbi12d 628	. . . . . 6 |- (y = {x} -> ((y e. Fin /\ x = |^|y) <-> ({x} e. Fin /\ x = |^|{x})))
10	9	rcla4ev 1877	. . . . 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 700	. . 3 |- ({x} e. P~A -> E.y e. P~ A(y e. Fin /\ x = |^|y))
13	3	snelpw 2752	. . 3 |- (x e. A <-> {x} e. P~A)
14		df-rex 1650	. . . 4 |- (E.y e. P~ A(y e. Fin /\ x = |^|y) <-> E.y(y e. P~A /\ (y e. Fin /\ x = |^|y)))
15		visset 1813	. . . . . . . 8 |- y e. V
16	15	elpw 2404	. . . . . . 7 |- (y e. P~A <-> y (_ A)
17	16	anbi1i 481	. . . . . 6 |- ((y e. P~A /\ (y e. Fin /\ x = |^|y)) <-> (y (_ A /\ (y e. Fin /\ x = |^|y)))
18		3anass 779	. . . . . 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 1051	. . . 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 988	1 |- A (_ {x | E.y(y (_ A /\ y e. Fin /\ x = |^|y)}

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  E.wex 980  {cab 1463  E.wrex 1646   (_ wss 2047  P~cpw 2401  {csn 2409  |^|cint 2533  Fincfn 4367

This theorem is referenced by:  abfi2 10490  efilcp 10572

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-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  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-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  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-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-1o 4133  df-en 4368  df-fin 4371

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