Theorem funchomf 49601

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 2741	. . . . . 6 ⊢ (Base‘𝐴) = (Base‘𝐴)
2		eqid 2741	. . . . . 6 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
3		eqid 2741	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		funchomf.1	. . . . . . 7 ⊢ (𝜑 → 𝐹(𝐴 Func 𝐶)𝐺)
5	4	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝐹(𝐴 Func 𝐶)𝐺)
6		simprl 777	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝐴))
7		simprr 779	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐴))
8	1, 2, 3, 5, 6, 7	funcf2 17830	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐴)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐶)(𝐹‘𝑦)))
9	8	ffnd 6660	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦) Fn (𝑥(Hom ‘𝐴)𝑦))
10		eqid 2741	. . . . . 6 ⊢ (Base‘𝐵) = (Base‘𝐵)
11		eqid 2741	. . . . . 6 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
12		eqid 2741	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
13		funchomf.2	. . . . . . 7 ⊢ (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺)
14	13	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝐹(𝐵 Func 𝐷)𝐺)
15		eqid 2741	. . . . . . . . . . 11 ⊢ (Base‘𝐶) = (Base‘𝐶)
16	1, 15, 4	funcf1 17828	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(Base‘𝐴)⟶(Base‘𝐶))
17	16	ffnd 6660	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐴))
18		eqid 2741	. . . . . . . . . . 11 ⊢ (Base‘𝐷) = (Base‘𝐷)
19	10, 18, 13	funcf1 17828	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(Base‘𝐵)⟶(Base‘𝐷))
20	19	ffnd 6660	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐵))
21		fndmu 6596	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝐴) ∧ 𝐹 Fn (Base‘𝐵)) → (Base‘𝐴) = (Base‘𝐵))
22	17, 20, 21	syl2anc 591	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
23	22	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (Base‘𝐴) = (Base‘𝐵))
24	6, 23	eleqtrd 2843	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝐵))
25	7, 23	eleqtrd 2843	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐵))
26	10, 11, 12, 14, 24, 25	funcf2 17830	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐵)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
27	26	ffnd 6660	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦) Fn (𝑥(Hom ‘𝐵)𝑦))
28		fndmu 6596	. . . 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 591	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
30	29	ralrimivva 3184	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)(𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
31		eqidd 2742	. . 3 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐴))
32	2, 11, 31, 22	homfeq 17655	. 2 ⊢ (𝜑 → ((Homf ‘𝐴) = (Homf ‘𝐵) ↔ ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)(𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦)))
33	30, 32	mpbird 259	1 ⊢ (𝜑 → (Homf ‘𝐴) = (Homf ‘𝐵))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  ∀wral 3055   class class class wbr 5075   Fn wfn 6484  ‘cfv 6489  (class class class)co 7360  Basecbs 17174  Hom chom 17226  Hom_f chomf 17627   Func cfunc 17816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8769  df-ixp 8840  df-homf 17631  df-func 17820

This theorem is referenced by:  idfu1stalem  49604  idfu2nda  49607  fthcomf  49661