Theorem comffn 17414

Description: The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
comfffn.o	⊢ 𝑂 = (compf‘𝐶)
comfffn.b	⊢ 𝐵 = (Base‘𝐶)
comffn.h	⊢ 𝐻 = (Hom ‘𝐶)
comffn.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
comffn.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
comffn.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
comffn	⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)))

Proof of Theorem comffn

Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2738	. . 3 ⊢ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓))
2		ovex 7308	. . 3 ⊢ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓) ∈ V
3	1, 2	fnmpoi 7910	. 2 ⊢ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))
4		comfffn.o	. . . 4 ⊢ 𝑂 = (compf‘𝐶)
5		comfffn.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
6		comffn.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
7		eqid 2738	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
8		comffn.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
9		comffn.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
10		comffn.z	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
11	4, 5, 6, 7, 8, 9, 10	comffval 17408	. . 3 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)))
12	11	fneq1d 6526	. 2 ⊢ (𝜑 → ((⟨𝑋, 𝑌⟩𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)) ↔ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))))
13	3, 12	mpbiri 257	1 ⊢ (𝜑 → (⟨𝑋, 𝑌⟩𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  ⟨cop 4567   × cxp 5587   Fn wfn 6428  ‘cfv 6433  (class class class)co 7275   ∈ cmpo 7277  Basecbs 16912  Hom chom 16973  compcco 16974  comp_fccomf 17376

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-comf 17380

This theorem is referenced by: (None)