Theorem ixpf 4346

Description: A member of an infinite Cartesian product maps to the indexed union of the product argument. Remark in [Enderton] p. 54.

Assertion

Ref	Expression
ixpf	|- (F e. X_x e. A B -> F:A-->U_x e. A B)

Distinct variable groups:   x,A   x,F

Proof of Theorem ixpf

Step	Hyp	Ref	Expression
1		elixp2 4339	. 2 |- (F e. X_x e. A B <-> (F e. V /\ F Fn A /\ A.x e. A (F` x) e. B))
2		ssiun2 2588	. . . . . . 7 |- (x e. A -> B (_ U_x e. A B)
3	2	sseld 2063	. . . . . 6 |- (x e. A -> ((F` x) e. B -> (F` x) e. U_x e. A B))
4	3	r19.20i 1701	. . . . 5 |- (A.x e. A (F` x) e. B -> A.x e. A (F` x) e. U_x e. A B)
5	4	anim2i 335	. . . 4 |- ((F Fn A /\ A.x e. A (F` x) e. B) -> (F Fn A /\ A.x e. A (F` x) e. U_x e. A B))
6		ax-17 969	. . . . 5 |- (y e. A -> A.x y e. A)
7		hbiu1 2579	. . . . 5 |- (y e. U_x e. A B -> A.x y e. U_x e. A B)
8		ax-17 969	. . . . 5 |- (y e. F -> A.x y e. F)
9	6, 7, 8	ffnfvf 3820	. . . 4 |- (F:A-->U_x e. A B <-> (F Fn A /\ A.x e. A (F` x) e. U_x e. A B))
10	5, 9	sylibr 200	. . 3 |- ((F Fn A /\ A.x e. A (F` x) e. B) -> F:A-->U_x e. A B)
11	10	3adant1 796	. 2 |- ((F e. V /\ F Fn A /\ A.x e. A (F` x) e. B) -> F:A-->U_x e. A B)
12	1, 11	sylbi 199	1 |- (F e. X_x e. A B -> F:A-->U_x e. A B)

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   e. wcel 956  A.wral 1642  Vcvv 1807  U_ciun 2561   Fn wfn 3172  -->wf 3173  ` cfv 3177  X_cixp 4337

This theorem is referenced by:  uniixp 4347  ixpssmap 4353

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-iun 2563  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-fv 3193  df-ixp 4338

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