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Theorem comffn 16970
 Description: The functionalized composition operation is a function. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfffn.o 𝑂 = (compf𝐶)
comfffn.b 𝐵 = (Base‘𝐶)
comffn.h 𝐻 = (Hom ‘𝐶)
comffn.x (𝜑𝑋𝐵)
comffn.y (𝜑𝑌𝐵)
comffn.z (𝜑𝑍𝐵)
Assertion
Ref Expression
comffn (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)))

Proof of Theorem comffn
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2798 . . 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 7169 . . 3 (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓) ∈ V
31, 2fnmpoi 7753 . 2 (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))
4 comfffn.o . . . 4 𝑂 = (compf𝐶)
5 comfffn.b . . . 4 𝐵 = (Base‘𝐶)
6 comffn.h . . . 4 𝐻 = (Hom ‘𝐶)
7 eqid 2798 . . . 4 (comp‘𝐶) = (comp‘𝐶)
8 comffn.x . . . 4 (𝜑𝑋𝐵)
9 comffn.y . . . 4 (𝜑𝑌𝐵)
10 comffn.z . . . 4 (𝜑𝑍𝐵)
114, 5, 6, 7, 8, 9, 10comffval 16964 . . 3 (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)))
1211fneq1d 6417 . 2 (𝜑 → ((⟨𝑋, 𝑌𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)) ↔ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))))
133, 12mpbiri 261 1 (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ⟨cop 4531   × cxp 5518   Fn wfn 6320  ‘cfv 6325  (class class class)co 7136   ∈ cmpo 7138  Basecbs 16478  Hom chom 16571  compcco 16572  compfccomf 16933 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pow 5232  ax-pr 5296  ax-un 7444 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ov 7139  df-oprab 7140  df-mpo 7141  df-1st 7674  df-2nd 7675  df-comf 16937 This theorem is referenced by: (None)
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