Theorem ixpconstg 7917

Description: Infinite Cartesian product of a constant B. (Contributed by Mario Carneiro, 11-Jan-2015.)

Assertion

Ref	Expression
ixpconstg	|- ( ( A e. V /\ B e. W ) -> X_ x e. A B = ( B ^m A ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   V( x)   W( x)

Proof of Theorem ixpconstg

Dummy variable f is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mapvalg 7867	. . 3 |- ( ( B e. W /\ A e. V ) -> ( B ^m A ) = { f | f : A --> B } )
2		vex 3203	. . . . 5 |- f e. _V
3	2	elixpconst 7916	. . . 4 |- ( f e. X_ x e. A B <-> f : A --> B )
4	3	abbi2i 2738	. . 3 |- X_ x e. A B = { f | f : A --> B }
5	1, 4	syl6reqr 2675	. 2 |- ( ( B e. W /\ A e. V ) -> X_ x e. A B = ( B ^m A ) )
6	5	ancoms 469	1 |- ( ( A e. V /\ B e. W ) -> X_ x e. A B = ( B ^m A ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   -->wf 5884  (class class class)co 6650   ^m cmap 7857   X_cixp 7908

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ixp 7909

This theorem is referenced by:  ixpconst  7918  mapsnf1o  7949  prdshom  16127  pwsbas  16147  frlmip  20117  pttoponconst  21400  xkoptsub  21457  xkopt  21458  tmdgsum2  21900  rrxip  23178  ovnlecvr2  40824