Theorem funchomf 49794

Description: Source categories of a functor have the same set of objects and morphisms. (Contributed by Zhi Wang, 10-Nov-2025.)

Hypotheses

Ref	Expression
funchomf.1	⊢ (𝜑 → 𝐹(𝐴 Func 𝐶)𝐺)
funchomf.2	⊢ (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺)

Assertion

Ref	Expression
funchomf	⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))

Proof of Theorem funchomf

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

Step	Hyp	Ref	Expression
1		eqid 2769	. . . . . 6 ⊢ (Base‘𝐴) = (Base‘𝐴)
2		eqid 2769	. . . . . 6 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
3		eqid 2769	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		funchomf.1	. . . . . . 7 ⊢ (𝜑 → 𝐹(𝐴 Func 𝐶)𝐺)
5	4	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝐹(𝐴 Func 𝐶)𝐺)
6		simprl 782	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝐴))
7		simprr 784	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐴))
8	1, 2, 3, 5, 6, 7	funcf2 17925	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐴)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐶)(𝐹‘𝑦)))
9	8	ffnd 6707	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦) Fn (𝑥(Hom ‘𝐴)𝑦))
10		eqid 2769	. . . . . 6 ⊢ (Base‘𝐵) = (Base‘𝐵)
11		eqid 2769	. . . . . 6 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
12		eqid 2769	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
13		funchomf.2	. . . . . . 7 ⊢ (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺)
14	13	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝐹(𝐵 Func 𝐷)𝐺)
15		eqid 2769	. . . . . . . . . . 11 ⊢ (Base‘𝐶) = (Base‘𝐶)
16	1, 15, 4	funcf1 17923	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(Base‘𝐴)⟶(Base‘𝐶))
17	16	ffnd 6707	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐴))
18		eqid 2769	. . . . . . . . . . 11 ⊢ (Base‘𝐷) = (Base‘𝐷)
19	10, 18, 13	funcf1 17923	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(Base‘𝐵)⟶(Base‘𝐷))
20	19	ffnd 6707	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐵))
21		fndmu 6643	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝐴) ∧ 𝐹 Fn (Base‘𝐵)) → (Base‘𝐴) = (Base‘𝐵))
22	17, 20, 21	syl2anc 595	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
23	22	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (Base‘𝐴) = (Base‘𝐵))
24	6, 23	eleqtrd 2871	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝐵))
25	7, 23	eleqtrd 2871	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐵))
26	10, 11, 12, 14, 24, 25	funcf2 17925	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐵)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
27	26	ffnd 6707	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦) Fn (𝑥(Hom ‘𝐵)𝑦))
28		fndmu 6643	. . . 4 ⊢ (((𝑥𝐺𝑦) Fn (𝑥(Hom ‘𝐴)𝑦) ∧ (𝑥𝐺𝑦) Fn (𝑥(Hom ‘𝐵)𝑦)) → (𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
29	9, 27, 28	syl2anc 595	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
30	29	ralrimivva 3214	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)(𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
31		eqidd 2770	. . 3 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐴))
32	2, 11, 31, 22	homfeq 17750	. 2 ⊢ (𝜑 → ((Homf ‘𝐴) = (Homf ‘𝐵) ↔ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)(𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦)))
33	30, 32	mpbird 260	1 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1567   ∈ wcel 2149  ∀wral 3085   class class class wbr 5113   Fn wfn 6532  ‘cfv 6537  (class class class)co 7411  Basecbs 17269  Hom chom 17321  Hom_f chomf 17722   Func cfunc 17911

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-iun 4962  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7414  df-oprab 7415  df-mpo 7416  df-1st 7986  df-2nd 7987  df-map 8826  df-ixp 8896  df-homf 17726  df-func 17915

This theorem is referenced by:  idfu1stalem  49797  idfu2nda  49800  fthcomf  49854