Theorem funchomf 49005

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 2735	. . . . . 6 ⊢ (Base‘𝐴) = (Base‘𝐴)
2		eqid 2735	. . . . . 6 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
3		eqid 2735	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		funchomf.1	. . . . . . 7 ⊢ (𝜑 → 𝐹(𝐴 Func 𝐶)𝐺)
5	4	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝐹(𝐴 Func 𝐶)𝐺)
6		simprl 770	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝐴))
7		simprr 772	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐴))
8	1, 2, 3, 5, 6, 7	funcf2 17879	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐴)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐶)(𝐹‘𝑦)))
9	8	ffnd 6706	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦) Fn (𝑥(Hom ‘𝐴)𝑦))
10		eqid 2735	. . . . . 6 ⊢ (Base‘𝐵) = (Base‘𝐵)
11		eqid 2735	. . . . . 6 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
12		eqid 2735	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
13		funchomf.2	. . . . . . 7 ⊢ (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺)
14	13	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝐹(𝐵 Func 𝐷)𝐺)
15		eqid 2735	. . . . . . . . . . 11 ⊢ (Base‘𝐶) = (Base‘𝐶)
16	1, 15, 4	funcf1 17877	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(Base‘𝐴)⟶(Base‘𝐶))
17	16	ffnd 6706	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐴))
18		eqid 2735	. . . . . . . . . . 11 ⊢ (Base‘𝐷) = (Base‘𝐷)
19	10, 18, 13	funcf1 17877	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(Base‘𝐵)⟶(Base‘𝐷))
20	19	ffnd 6706	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐵))
21		fndmu 6644	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝐴) ∧ 𝐹 Fn (Base‘𝐵)) → (Base‘𝐴) = (Base‘𝐵))
22	17, 20, 21	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
23	22	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (Base‘𝐴) = (Base‘𝐵))
24	6, 23	eleqtrd 2836	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝐵))
25	7, 23	eleqtrd 2836	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐵))
26	10, 11, 12, 14, 24, 25	funcf2 17879	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐵)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
27	26	ffnd 6706	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦) Fn (𝑥(Hom ‘𝐵)𝑦))
28		fndmu 6644	. . . 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 584	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
30	29	ralrimivva 3187	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)(𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
31		eqidd 2736	. . 3 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐴))
32	2, 11, 31, 22	homfeq 17704	. 2 ⊢ (𝜑 → ((Homf ‘𝐴) = (Homf ‘𝐵) ↔ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)(𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦)))
33	30, 32	mpbird 257	1 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2108  ∀wral 3051   class class class wbr 5119   Fn wfn 6525  ‘cfv 6530  (class class class)co 7403  Basecbs 17226  Hom chom 17280  Hom_f chomf 17676   Func cfunc 17865

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 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-ov 7406  df-oprab 7407  df-mpo 7408  df-1st 7986  df-2nd 7987  df-map 8840  df-ixp 8910  df-homf 17680  df-func 17869

This theorem is referenced by:  idfu1stalem  49007  idfu2nda  49010  fthcomf  49045