Theorem uobeq 49849

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 17909	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
8		fullfunc 17955	. . . . . 6 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
9	8, 6	sselid 3937	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
10		1st2nd 8024	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
11	7, 9, 10	sylancr 598	. . . 4 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
12		uobeq.i	. . . . . 6 ⊢ 𝐼 = (idfunc‘𝐷)
13	9	func1st2nd 49705	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
14		uobeq.l	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (𝐸 Func 𝐷))
15	14	func1st2nd 49705	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐿)(𝐸 Func 𝐷)(2nd ‘𝐿))
16	9, 14	cofu1st2nd 49721	. . . . . . 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 2802	. . . . . 6 ⊢ (𝜑 → (⟨(1st ‘𝐿), (2nd ‘𝐿)⟩ ∘func ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩) = 𝐼)
19	12, 13, 15, 18	cofidfth 49791	. . . . 5 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Faith 𝐸)(2nd ‘𝐾))
20		df-br 5106	. . . . 5 ⊢ ((1st ‘𝐾)(𝐷 Faith 𝐸)(2nd ‘𝐾) ↔ ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ ∈ (𝐷 Faith 𝐸))
21	19, 20	sylib 221	. . . 4 ⊢ (𝜑 → ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩ ∈ (𝐷 Faith 𝐸))
22	11, 21	eqeltrd 2865	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Faith 𝐸))
23	6, 22	elind 4155	. 2 ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
24	1, 2, 3, 4, 5, 23	uobffth 49847	1 ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  ⟨cop 4591   class class class wbr 5105  dom cdm 5652  Rel wrel 5657  ‘cfv 6525  (class class class)co 7400  1^st c1st 7972  2^nd c2nd 7973  Basecbs 17259   Func cfunc 17901  id_funccidfu 17902   ∘_func ccofu 17903   Full cful 17951   Faith cfth 17952   UP cup 49802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-map 8814  df-ixp 8884  df-cat 17714  df-cid 17715  df-func 17905  df-idfu 17906  df-cofu 17907  df-full 17953  df-fth 17954  df-up 49803

This theorem is referenced by:  uobeq2  50030