Theorem funchomf 50088

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 2760	. . . . . 6 ⊢ (Base‘𝐴) = (Base‘𝐴)
2		eqid 2760	. . . . . 6 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
3		eqid 2760	. . . . . 6 ⊢ (Hom ‘𝐶) = (Hom ‘𝐶)
4		funchomf.1	. . . . . . 7 ⊢ (𝜑 → 𝐹(𝐴 Func 𝐶)𝐺)
5	4	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝐹(𝐴 Func 𝐶)𝐺)
6		simprl 783	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝐴))
7		simprr 785	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐴))
8	1, 2, 3, 5, 6, 7	funcf2 17982	. . . . 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 2760	. . . . . 6 ⊢ (Base‘𝐵) = (Base‘𝐵)
11		eqid 2760	. . . . . 6 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
12		eqid 2760	. . . . . 6 ⊢ (Hom ‘𝐷) = (Hom ‘𝐷)
13		funchomf.2	. . . . . . 7 ⊢ (𝜑 → 𝐹(𝐵 Func 𝐷)𝐺)
14	13	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝐹(𝐵 Func 𝐷)𝐺)
15		eqid 2760	. . . . . . . . . . 11 ⊢ (Base‘𝐶) = (Base‘𝐶)
16	1, 15, 4	funcf1 17980	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(Base‘𝐴)⟶(Base‘𝐶))
17	16	ffnd 6706	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐴))
18		eqid 2760	. . . . . . . . . . 11 ⊢ (Base‘𝐷) = (Base‘𝐷)
19	10, 18, 13	funcf1 17980	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(Base‘𝐵)⟶(Base‘𝐷))
20	19	ffnd 6706	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐵))
21		fndmu 6642	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝐴) ∧ 𝐹 Fn (Base‘𝐵)) → (Base‘𝐴) = (Base‘𝐵))
22	17, 20, 21	syl2anc 596	. . . . . . . 8 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐵))
23	22	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (Base‘𝐴) = (Base‘𝐵))
24	6, 23	eleqtrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝐵))
25	7, 23	eleqtrd 2862	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐵))
26	10, 11, 12, 14, 24, 25	funcf2 17982	. . . . 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 6642	. . . 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 596	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
30	29	ralrimivva 3205	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)(𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
31		eqidd 2761	. . 3 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐴))
32	2, 11, 31, 22	homfeq 17807	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076   class class class wbr 5103   Fn wfn 6530  ‘cfv 6535  (class class class)co 7416  Basecbs 17326  Hom chom 17378  Hom_f chomf 17779   Func cfunc 17968

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7742

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 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  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 6491  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7419  df-oprab 7420  df-mpo 7421  df-1st 7992  df-2nd 7993  df-map 8835  df-ixp 8912  df-homf 17783  df-func 17972

This theorem is used by:  idfu1stalem  50091  idfu2nda  50094  fthcomf  50148