MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  comffn Structured version   Visualization version   GIF version

Theorem comffn 16297
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 2621 . . 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 6638 . . 3 (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓) ∈ V
31, 2fnmpt2i 7191 . 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 2621 . . . 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 16291 . . 3 (𝜑 → (⟨𝑋, 𝑌𝑂𝑍) = (𝑔 ∈ (𝑌𝐻𝑍), 𝑓 ∈ (𝑋𝐻𝑌) ↦ (𝑔(⟨𝑋, 𝑌⟩(comp‘𝐶)𝑍)𝑓)))
1211fneq1d 5944 . 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 1480  wcel 1987  cop 4159   × cxp 5077   Fn wfn 5847  cfv 5852  (class class class)co 6610  cmpt2 6612  Basecbs 15792  Hom chom 15884  compcco 15885  compfccomf 16260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-comf 16264
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator