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Theorem comffn 16572
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 2771 . . 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 6823 . . 3 (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓) ∈ V
31, 2fnmpt2i 7389 . 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 2771 . . . 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 16566 . . 3 (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)))
1211fneq1d 6121 . 2 (𝜑 → ((⟨𝑋, 𝑌𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)) ↔ (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌))))
133, 12mpbiri 248 1 (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) Fn ((𝑌𝐻𝑍) × (𝑋𝐻𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cop 4322   × cxp 5247   Fn wfn 6026  cfv 6031  (class class class)co 6793  cmpt2 6795  Basecbs 16064  Hom chom 16160  compcco 16161  compfccomf 16535
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-rep 4904  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-reu 3068  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-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-comf 16539
This theorem is referenced by: (None)
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