Theorem uobeq 49209

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 17824	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
8		fullfunc 17870	. . . . . 6 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
9	8, 6	sselid 3944	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
10		1st2nd 8018	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
11	7, 9, 10	sylancr 587	. . . 4 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
12		uobeq.i	. . . . . 6 ⊢ 𝐼 = (idfunc‘𝐷)
13	9	func1st2nd 49065	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
14		uobeq.l	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (𝐸 Func 𝐷))
15	14	func1st2nd 49065	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐿)(𝐸 Func 𝐷)(2nd ‘𝐿))
16	9, 14	cofu1st2nd 49081	. . . . . . 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 2766	. . . . . 6 ⊢ (𝜑 → (⟨(1st ‘𝐿), (2nd ‘𝐿)⟩ ∘func ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩) = 𝐼)
19	12, 13, 15, 18	cofidfth 49151	. . . . 5 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Faith 𝐸)(2nd ‘𝐾))
20		df-br 5108	. . . . 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 2828	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Faith 𝐸))
23	6, 22	elind 4163	. 2 ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
24	1, 2, 3, 4, 5, 23	uobffth 49207	1 ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⟨cop 4595   class class class wbr 5107  dom cdm 5638  Rel wrel 5643  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  Basecbs 17179   Func cfunc 17816  id_funccidfu 17817   ∘_func ccofu 17818   Full cful 17866   Faith cfth 17867   UP cup 49162

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-map 8801  df-ixp 8871  df-cat 17629  df-cid 17630  df-func 17820  df-idfu 17821  df-cofu 17822  df-full 17868  df-fth 17869  df-up 49163

This theorem is referenced by:  uobeq2  49390