Theorem funchomf 49409

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 2737	. . . . . 6 ⊢ (Base‘𝐴) = (Base‘𝐴)
2		eqid 2737	. . . . . 6 ⊢ (Hom ‘𝐴) = (Hom ‘𝐴)
3		eqid 2737	. . . . . 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 771	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝐴))
7		simprr 773	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐴))
8	1, 2, 3, 5, 6, 7	funcf2 17796	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐴)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐶)(𝐹‘𝑦)))
9	8	ffnd 6664	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦) Fn (𝑥(Hom ‘𝐴)𝑦))
10		eqid 2737	. . . . . 6 ⊢ (Base‘𝐵) = (Base‘𝐵)
11		eqid 2737	. . . . . 6 ⊢ (Hom ‘𝐵) = (Hom ‘𝐵)
12		eqid 2737	. . . . . 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 2737	. . . . . . . . . . 11 ⊢ (Base‘𝐶) = (Base‘𝐶)
16	1, 15, 4	funcf1 17794	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(Base‘𝐴)⟶(Base‘𝐶))
17	16	ffnd 6664	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐴))
18		eqid 2737	. . . . . . . . . . 11 ⊢ (Base‘𝐷) = (Base‘𝐷)
19	10, 18, 13	funcf1 17794	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:(Base‘𝐵)⟶(Base‘𝐷))
20	19	ffnd 6664	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 Fn (Base‘𝐵))
21		fndmu 6600	. . . . . . . . 9 ⊢ ((𝐹 Fn (Base‘𝐴) ∧ 𝐹 Fn (Base‘𝐵)) → (Base‘𝐴) = (Base‘𝐵))
22	17, 20, 21	syl2anc 585	. . . . . . . 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 2839	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑥 ∈ (Base‘𝐵))
25	7, 23	eleqtrd 2839	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → 𝑦 ∈ (Base‘𝐵))
26	10, 11, 12, 14, 24, 25	funcf2 17796	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦):(𝑥(Hom ‘𝐵)𝑦)⟶((𝐹‘𝑥)(Hom ‘𝐷)(𝐹‘𝑦)))
27	26	ffnd 6664	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥𝐺𝑦) Fn (𝑥(Hom ‘𝐵)𝑦))
28		fndmu 6600	. . . 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 585	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (Base‘𝐴) ∧ 𝑦 ∈ (Base‘𝐴))) → (𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
30	29	ralrimivva 3180	. 2 ⊢ (𝜑 → ∀𝑥 ∈ (Base‘𝐴)∀𝑦 ∈ (Base‘𝐴)(𝑥(Hom ‘𝐴)𝑦) = (𝑥(Hom ‘𝐵)𝑦))
31		eqidd 2738	. . 3 ⊢ (𝜑 → (Base‘𝐴) = (Base‘𝐴))
32	2, 11, 31, 22	homfeq 17621	. 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 1542   ∈ wcel 2114  ∀wral 3052   class class class wbr 5099   Fn wfn 6488  ‘cfv 6493  (class class class)co 7360  Basecbs 17140  Hom chom 17192  Hom_f chomf 17593   Func cfunc 17782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-map 8769  df-ixp 8840  df-homf 17597  df-func 17786

This theorem is referenced by:  idfu1stalem  49412  idfu2nda  49415  fthcomf  49469