Theorem ipffn 20567

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 × 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 2736	. . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
3		ipffn.2	. . 3 ⊢ , = (·if‘𝑊)
4	1, 2, 3	ipffval 20564	. 2 ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦))
5		ovex 7224	. 2 ⊢ (𝑥(·𝑖‘𝑊)𝑦) ∈ V
6	4, 5	fnmpoi 7818	1 ⊢ , Fn (𝑉 × 𝑉)

Syntax hints:   = wceq 1543   × cxp 5534   Fn wfn 6353  ‘cfv 6358  (class class class)co 7191  Basecbs 16666  ·_𝑖cip 16754  ·_ifcipf 20541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-id 5440  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-fv 6366  df-ov 7194  df-oprab 7195  df-mpo 7196  df-1st 7739  df-2nd 7740  df-ipf 20543

This theorem is referenced by: (None)