Theorem uobeq 49695

Description: If a full functor (in fact, a full embedding) is a section of a functor (surjective on objects), then the sets of universal objects are equal. (Contributed by Zhi Wang, 17-Nov-2025.)

Hypotheses

Ref	Expression
uobffth.b	⊢ 𝐵 = (Base‘𝐷)
uobffth.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
uobffth.f	⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
uobffth.g	⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)
uobffth.y	⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)
uobeq.i	⊢ 𝐼 = (idfunc‘𝐷)
uobeq.k	⊢ (𝜑 → 𝐾 ∈ (𝐷 Full 𝐸))
uobeq.n	⊢ (𝜑 → (𝐿 ∘func 𝐾) = 𝐼)
uobeq.l	⊢ (𝜑 → 𝐿 ∈ (𝐸 Func 𝐷))

Assertion

Ref	Expression
uobeq	⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))

Proof of Theorem uobeq

Step	Hyp	Ref	Expression
1		uobffth.b	. 2 ⊢ 𝐵 = (Base‘𝐷)
2		uobffth.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		uobffth.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝐶 Func 𝐷))
4		uobffth.g	. 2 ⊢ (𝜑 → (𝐾 ∘func 𝐹) = 𝐺)
5		uobffth.y	. 2 ⊢ (𝜑 → ((1st ‘𝐾)‘𝑋) = 𝑌)
6		uobeq.k	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Full 𝐸))
7		relfunc 17829	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
8		fullfunc 17875	. . . . . 6 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
9	8, 6	sselid 3919	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
10		1st2nd 7992	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
11	7, 9, 10	sylancr 588	. . . 4 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
12		uobeq.i	. . . . . 6 ⊢ 𝐼 = (idfunc‘𝐷)
13	9	func1st2nd 49551	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
14		uobeq.l	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (𝐸 Func 𝐷))
15	14	func1st2nd 49551	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐿)(𝐸 Func 𝐷)(2nd ‘𝐿))
16	9, 14	cofu1st2nd 49567	. . . . . . 7 ⊢ (𝜑 → (𝐿 ∘func 𝐾) = (⟨(1st ‘𝐿), (2nd ‘𝐿)⟩ ∘func ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩))
17		uobeq.n	. . . . . . 7 ⊢ (𝜑 → (𝐿 ∘func 𝐾) = 𝐼)
18	16, 17	eqtr3d 2773	. . . . . 6 ⊢ (𝜑 → (⟨(1st ‘𝐿), (2nd ‘𝐿)⟩ ∘func ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩) = 𝐼)
19	12, 13, 15, 18	cofidfth 49637	. . . . 5 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Faith 𝐸)(2nd ‘𝐾))
20		df-br 5086	. . . . 5 ⊢ ((1st ‘𝐾)(𝐷 Faith 𝐸)(2nd ‘𝐾) ↔ ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ ∈ (𝐷 Faith 𝐸))
21	19, 20	sylib 218	. . . 4 ⊢ (𝜑 → ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ ∈ (𝐷 Faith 𝐸))
22	11, 21	eqeltrd 2836	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Faith 𝐸))
23	6, 22	elind 4140	. 2 ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
24	1, 2, 3, 4, 5, 23	uobffth 49693	1 ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⟨cop 4573   class class class wbr 5085  dom cdm 5631  Rel wrel 5636  ‘cfv 6498  (class class class)co 7367  1^st c1st 7940  2^nd c2nd 7941  Basecbs 17179   Func cfunc 17821  id_funccidfu 17822   ∘_func ccofu 17823   Full cful 17871   Faith cfth 17872   UP cup 49648

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-1st 7942  df-2nd 7943  df-map 8775  df-ixp 8846  df-cat 17634  df-cid 17635  df-func 17825  df-idfu 17826  df-cofu 17827  df-full 17873  df-fth 17874  df-up 49649

This theorem is referenced by:  uobeq2  49876