Theorem uobeq 50329

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 18038	. . . . 5 ⊢ Rel (𝐷 Func 𝐸)
8		fullfunc 18084	. . . . . 6 ⊢ (𝐷 Full 𝐸) ⊆ (𝐷 Func 𝐸)
9	8, 6	sselid 3927	. . . . 5 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Func 𝐸))
10		1st2nd 8051	. . . . 5 ⊢ ((Rel (𝐷 Func 𝐸) ∧ 𝐾 ∈ (𝐷 Func 𝐸)) → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
11	7, 9, 10	sylancr 599	. . . 4 ⊢ (𝜑 → 𝐾 = ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩)
12		uobeq.i	. . . . . 6 ⊢ 𝐼 = (idfunc‘𝐷)
13	9	func1st2nd 50185	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Func 𝐸)(2nd ‘𝐾))
14		uobeq.l	. . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ (𝐸 Func 𝐷))
15	14	func1st2nd 50185	. . . . . 6 ⊢ (𝜑 → (1st ‘𝐿)(𝐸 Func 𝐷)(2nd ‘𝐿))
16	9, 14	cofu1st2nd 50201	. . . . . . 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 2798	. . . . . 6 ⊢ (𝜑 → (⟨(1st ‘𝐿), (2nd ‘𝐿)⟩ ∘func ⟨(1st ‘𝐾), (2nd ‘𝐾)⟩) = 𝐼)
19	12, 13, 15, 18	cofidfth 50271	. . . . 5 ⊢ (𝜑 → (1st ‘𝐾)(𝐷 Faith 𝐸)(2nd ‘𝐾))
20		df-br 5104	. . . . 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 2861	. . 3 ⊢ (𝜑 → 𝐾 ∈ (𝐷 Faith 𝐸))
23	6, 22	elind 4146	. 2 ⊢ (𝜑 → 𝐾 ∈ ((𝐷 Full 𝐸) ∩ (𝐷 Faith 𝐸)))
24	1, 2, 3, 4, 5, 23	uobffth 50327	1 ⊢ (𝜑 → dom (𝐹(𝐶 UP 𝐷)𝑋) = dom (𝐺(𝐶 UP 𝐸)𝑌))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ⟨cop 4590   class class class wbr 5103  dom cdm 5651  Rel wrel 5656  ‘cfv 6539  (class class class)co 7421  1^st c1st 8000  2^nd c2nd 8001  Basecbs 17388   Func cfunc 18030  id_funccidfu 18031   ∘_func ccofu 18032   Full cful 18080   Faith cfth 18081   UP cup 50282

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7752

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3900  df-un 3902  df-in 3904  df-ss 3915  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 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-riota 7378  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-map 8850  df-ixp 8927  df-cat 17843  df-cid 17844  df-func 18034  df-idfu 18035  df-cofu 18036  df-full 18082  df-fth 18083  df-up 50283

This theorem is used by:  uobeq2  50510