Theorem ipffn 19996

Description: The inner product operation is a function. (Contributed by Mario Carneiro, 20-Sep-2015.)

Hypotheses

Ref	Expression
ipffn.1	|- V = ( Base ` W )
ipffn.2	|- ., = ( .if ` W )

Assertion

Ref	Expression
ipffn	|- ., Fn ( V X. V )

Proof of Theorem ipffn

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ipffn.1	. . 3 |- V = ( Base ` W )
2		eqid 2622	. . 3 |- ( .i ` W ) = ( .i ` W )
3		ipffn.2	. . 3 |- ., = ( .if ` W )
4	1, 2, 3	ipffval 19993	. 2 |- ., = ( x e. V , y e. V |-> ( x ( .i ` W ) y ) )
5		ovex 6678	. 2 |- ( x ( .i ` W ) y ) e. _V
6	4, 5	fnmpt2i 7239	1 |- ., Fn ( V X. V )

Syntax hints:   = wceq 1483   X. cxp 5112   Fn wfn 5883   ` cfv 5888  (class class class)co 6650   Basecbs 15857   .icip 15946   .ifcipf 19970

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-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  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-iun 4522  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-res 5126  df-ima 5127  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-1st 7168  df-2nd 7169  df-ipf 19972

This theorem is referenced by: (None)