Theorem elixp2 4349

Description: Membership in an infinite Cartesian product. See df-ixp 4348 for discussion of the notation.

Assertion

Ref	Expression
elixp2	|- (F e. X_x e. A B <-> (F e. V /\ F Fn A /\ A.x e. A (F` x) e. B))

Distinct variable group:   x,F

Proof of Theorem elixp2

Step	Hyp	Ref	Expression
1		fneq1 3582	. . . . 5 |- (f = F -> (f Fn A <-> F Fn A))
2		fveq1 3723	. . . . . . 7 |- (f = F -> (f` x) = (F` x))
3	2	eleq1d 1540	. . . . . 6 |- (f = F -> ((f` x) e. B <-> (F` x) e. B))
4	3	ralbidv 1663	. . . . 5 |- (f = F -> (A.x e. A (f` x) e. B <-> A.x e. A (F` x) e. B))
5	1, 4	anbi12d 628	. . . 4 |- (f = F -> ((f Fn A /\ A.x e. A (f` x) e. B) <-> (F Fn A /\ A.x e. A (F` x) e. B)))
6		df-ixp 4348	. . . 4 |- X_x e. A B = {f | (f Fn A /\ A.x e. A (f` x) e. B)}
7	5, 6	elab2g 1900	. . 3 |- (F e. V -> (F e. X_x e. A B <-> (F Fn A /\ A.x e. A (F` x) e. B)))
8	7	pm5.32i 645	. 2 |- ((F e. V /\ F e. X_x e. A B) <-> (F e. V /\ (F Fn A /\ A.x e. A (F` x) e. B)))
9		elisset 1817	. . 3 |- (F e. X_x e. A B -> F e. V)
10	9	pm4.71ri 638	. 2 |- (F e. X_x e. A B <-> (F e. V /\ F e. X_x e. A B))
11		3anass 779	. 2 |- ((F e. V /\ F Fn A /\ A.x e. A (F` x) e. B) <-> (F e. V /\ (F Fn A /\ A.x e. A (F` x) e. B)))
12	8, 10, 11	3bitr4 183	1 |- (F e. X_x e. A B <-> (F e. V /\ F Fn A /\ A.x e. A (F` x) e. B))

Syntax hints:   <-> wb 146   /\ wa 223   /\ w3a 775   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811   Fn wfn 3177  ` cfv 3182  X_cixp 4347

This theorem is referenced by:  elixp 4350  ixpf 4356

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-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

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-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-br 2620  df-opab 2667  df-id 2835  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-fv 3198  df-ixp 4348

Copyright terms: Public domain