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Theorem comfffn 17045
 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)
Assertion
Ref Expression
comfffn 𝑂 Fn ((𝐵 × 𝐵) × 𝐵)

Proof of Theorem comfffn
Dummy variables 𝑥 𝑦 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 comfffn.o . . 3 𝑂 = (compf𝐶)
2 comfffn.b . . 3 𝐵 = (Base‘𝐶)
3 eqid 2758 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
4 eqid 2758 . . 3 (comp‘𝐶) = (comp‘𝐶)
51, 2, 3, 4comfffval 17039 . 2 𝑂 = (𝑥 ∈ (𝐵 × 𝐵), 𝑦𝐵 ↦ (𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥(comp‘𝐶)𝑦)𝑓)))
6 ovex 7189 . . 3 ((2nd𝑥)(Hom ‘𝐶)𝑦) ∈ V
7 fvex 6676 . . 3 ((Hom ‘𝐶)‘𝑥) ∈ V
86, 7mpoex 7788 . 2 (𝑔 ∈ ((2nd𝑥)(Hom ‘𝐶)𝑦), 𝑓 ∈ ((Hom ‘𝐶)‘𝑥) ↦ (𝑔(𝑥(comp‘𝐶)𝑦)𝑓)) ∈ V
95, 8fnmpoi 7778 1 𝑂 Fn ((𝐵 × 𝐵) × 𝐵)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   × cxp 5526   Fn wfn 6335  ‘cfv 6340  (class class class)co 7156   ∈ cmpo 7158  2nd c2nd 7698  Basecbs 16554  Hom chom 16647  compcco 16648  compfccomf 17009 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 2729  ax-rep 5160  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5037  df-opab 5099  df-mpt 5117  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-f1 6345  df-fo 6346  df-f1o 6347  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7699  df-2nd 7700  df-comf 17013 This theorem is referenced by: (None)
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