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Theorem comffn 16963
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 2818 . . 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 7178 . . 3 (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓) ∈ V
31, 2fnmpoi 7757 . 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 2818 . . . 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 16957 . . 3 (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)))
1211fneq1d 6439 . 2 (𝜑 → ((⟨𝑋, 𝑌𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)) ↔ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))))
133, 12mpbiri 259 1 (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  cop 4563   × cxp 5546   Fn wfn 6343  cfv 6348  (class class class)co 7145  cmpo 7147  Basecbs 16471  Hom chom 16564  compcco 16565  compfccomf 16926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-comf 16930
This theorem is referenced by: (None)
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