Theorem ipffn 20213

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 2771	. . 3 ⊢ (·𝑖‘𝑊) = (·𝑖‘𝑊)
3		ipffn.2	. . 3 ⊢ , = (·if‘𝑊)
4	1, 2, 3	ipffval 20210	. 2 ⊢ , = (𝑥 ∈ 𝑉, 𝑦 ∈ 𝑉 ↦ (𝑥(·𝑖‘𝑊)𝑦))
5		ovex 6823	. 2 ⊢ (𝑥(·𝑖‘𝑊)𝑦) ∈ V
6	4, 5	fnmpt2i 7389	1 ⊢ , Fn (𝑉 × 𝑉)

Syntax hints:   = wceq 1631   × cxp 5247   Fn wfn 6026  ‘cfv 6031  (class class class)co 6793  Basecbs 16064  ·_𝑖cip 16154  ·_ifcipf 20187

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-ipf 20189

This theorem is referenced by: (None)