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

Theorem comfeqd 17750
Description: Condition for two categories with the same hom-sets to have the same composition. (Contributed by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
comfeqd.1 (𝜑 → (comp‘𝐶) = (comp‘𝐷))
comfeqd.2 (𝜑 → (Homf𝐶) = (Homf𝐷))
Assertion
Ref Expression
comfeqd (𝜑 → (compf𝐶) = (compf𝐷))

Proof of Theorem comfeqd
Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 comfeqd.1 . . . . . . . . 9 (𝜑 → (comp‘𝐶) = (comp‘𝐷))
21oveqd 7448 . . . . . . . 8 (𝜑 → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧) = (⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧))
32oveqd 7448 . . . . . . 7 (𝜑 → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))
43ralrimivw 3150 . . . . . 6 (𝜑 → ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))
54ralrimivw 3150 . . . . 5 (𝜑 → ∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))
65ralrimivw 3150 . . . 4 (𝜑 → ∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))
76ralrimivw 3150 . . 3 (𝜑 → ∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))
87ralrimivw 3150 . 2 (𝜑 → ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))
9 eqid 2737 . . 3 (comp‘𝐶) = (comp‘𝐶)
10 eqid 2737 . . 3 (comp‘𝐷) = (comp‘𝐷)
11 eqid 2737 . . 3 (Hom ‘𝐶) = (Hom ‘𝐶)
12 eqidd 2738 . . 3 (𝜑 → (Base‘𝐶) = (Base‘𝐶))
13 comfeqd.2 . . . 4 (𝜑 → (Homf𝐶) = (Homf𝐷))
1413homfeqbas 17739 . . 3 (𝜑 → (Base‘𝐶) = (Base‘𝐷))
159, 10, 11, 12, 14, 13comfeq 17749 . 2 (𝜑 → ((compf𝐶) = (compf𝐷) ↔ ∀𝑥 ∈ (Base‘𝐶)∀𝑦 ∈ (Base‘𝐶)∀𝑧 ∈ (Base‘𝐶)∀𝑓 ∈ (𝑥(Hom ‘𝐶)𝑦)∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓)))
168, 15mpbird 257 1 (𝜑 → (compf𝐶) = (compf𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wral 3061  cop 4632  cfv 6561  (class class class)co 7431  Basecbs 17247  Hom chom 17308  compcco 17309  Homf chomf 17709  compfccomf 17710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-homf 17713  df-comf 17714
This theorem is referenced by:  fullresc  17896
  Copyright terms: Public domain W3C validator