Theorem comfeqd 17907

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 → (Homf ‘𝐶) = (Homf ‘𝐷))

Assertion

Ref	Expression
comfeqd	⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))

Proof of Theorem comfeqd

Dummy variables 𝑓 𝑔 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		comfeqd.1	. . . . . . . . 9 ⊢ (𝜑 → (comp‘𝐶) = (comp‘𝐷))
2	1	oveqd 7457	. . . . . . . 8 ⊢ (𝜑 → (⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧) = (⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧))
3	2	oveqd 7457	. . . . . . 7 ⊢ (𝜑 → (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))
4	3	ralrimivw 3162	. . . . . 6 ⊢ (𝜑 → ∀𝑔 ∈ (𝑦(Hom ‘𝐶)𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝐶)𝑧)𝑓) = (𝑔(⟨𝑥, 𝑦⟩(comp‘𝐷)𝑧)𝑓))
5	4	ralrimivw 3162	. . . . 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))
6	5	ralrimivw 3162	. . . 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))
7	6	ralrimivw 3162	. . 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))
8	7	ralrimivw 3162	. 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 2763	. . 3 ⊢ (comp‘𝐶) = (comp‘𝐶)
10		eqid 2763	. . 3 ⊢ (comp‘𝐷) = (comp‘𝐷)
11		eqid 2763	. . 3 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
12		eqidd 2764	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐶))
13		comfeqd.2	. . . 4 ⊢ (𝜑 → (Homf ‘𝐶) = (Homf ‘𝐷))
14	13	homfeqbas 17896	. . 3 ⊢ (𝜑 → (Base‘𝐶) = (Base‘𝐷))
15	9, 10, 11, 12, 14, 13	comfeq 17906	. 2 ⊢ (𝜑 → ((compf‘𝐶) = (compf‘𝐷) ↔ ∀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)))
16	8, 15	mpbird 260	1 ⊢ (𝜑 → (compf‘𝐶) = (compf‘𝐷))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∀wral 3080  ⟨cop 4838  ‘cfv 6555  (class class class)co 7440  Basecbs 17413  Hom chom 17465  compcco 17466  Hom_f chomf 17866  comp_fccomf 17867

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-ov 7443  df-oprab 7444  df-mpo 7445  df-1st 8022  df-2nd 8023  df-homf 17870  df-comf 17871

This theorem is used by:  fullresc  18052