Theorem upfval3 50112

Description: Function value of the class of universal properties. (Contributed by Zhi Wang, 24-Sep-2025.)

Hypotheses

Ref	Expression
upfval.b	⊢ 𝐵 = (Base‘𝐷)
upfval.c	⊢ 𝐶 = (Base‘𝐸)
upfval.h	⊢ 𝐻 = (Hom ‘𝐷)
upfval.j	⊢ 𝐽 = (Hom ‘𝐸)
upfval.o	⊢ 𝑂 = (comp‘𝐸)
upfval2.w	⊢ (𝜑 → 𝑊 ∈ 𝐶)
upfval3.f	⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)

Assertion

Ref	Expression
upfval3	⊢ (𝜑 → (⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊) = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽(𝐹‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚))})

Distinct variable groups:   𝐵,𝑔,𝑘,𝑚,𝑥,𝑦   𝐶,𝑔,𝑘,𝑚,𝑥,𝑦   𝐷,𝑔,𝑘,𝑚,𝑥,𝑦   𝑔,𝐸,𝑘,𝑚,𝑥,𝑦   𝑔,𝐹,𝑘,𝑚,𝑥,𝑦   𝑔,𝐺,𝑘,𝑚,𝑥,𝑦   𝑔,𝐻,𝑘,𝑚,𝑥,𝑦   𝑔,𝐽,𝑘,𝑚,𝑥,𝑦   𝑔,𝑂,𝑘,𝑚,𝑥,𝑦   𝑔,𝑊,𝑘,𝑚,𝑥,𝑦   𝜑,𝑚,𝑥

Allowed substitution hints:   𝜑(𝑦, 𝑔, 𝑘)

Proof of Theorem upfval3

Step	Hyp	Ref	Expression
1		upfval.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
2		upfval.c	. . 3 ⊢ 𝐶 = (Base‘𝐸)
3		upfval.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐷)
4		upfval.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐸)
5		upfval.o	. . 3 ⊢ 𝑂 = (comp‘𝐸)
6		upfval2.w	. . 3 ⊢ (𝜑 → 𝑊 ∈ 𝐶)
7		upfval3.f	. . . 4 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
8		df-br 5108	. . . 4 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
9	7, 8	sylib 221	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
10	1, 2, 3, 4, 5, 6, 9	upfval2 50111	. 2 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊) = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)‘𝑘)(⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩𝑂((1st ‘⟨𝐹, 𝐺⟩)‘𝑦))𝑚))})
11		relfunc 17957	. . . . . . . . . . 11 ⊢ Rel (𝐷 Func 𝐸)
12	11	brrelex12i 5714	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
13		op1stg 8002	. . . . . . . . . 10 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
14	12, 13	syl 18	. . . . . . . . 9 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
15	14	fveq1d 6884	. . . . . . . 8 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥) = (𝐹‘𝑥))
16	15	oveq2d 7433	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)) = (𝑊𝐽(𝐹‘𝑥)))
17	16	eleq2d 2848	. . . . . 6 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑚 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)) ↔ 𝑚 ∈ (𝑊𝐽(𝐹‘𝑥))))
18	17	anbi2d 642	. . . . 5 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑥))) ↔ (𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽(𝐹‘𝑥)))))
19	14	fveq1d 6884	. . . . . . . 8 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑦) = (𝐹‘𝑦))
20	19	oveq2d 7433	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = (𝑊𝐽(𝐹‘𝑦)))
21	15	opeq2d 4843	. . . . . . . . . . 11 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩ = ⟨𝑊, (𝐹‘𝑥)⟩)
22	21, 19	oveq12d 7435	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩𝑂((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = (⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦)))
23		op2ndg 8003	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
24	12, 23	syl 18	. . . . . . . . . . . 12 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
25	24	oveqd 7434	. . . . . . . . . . 11 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦) = (𝑥𝐺𝑦))
26	25	fveq1d 6884	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ((𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)‘𝑘) = ((𝑥𝐺𝑦)‘𝑘))
27		eqidd 2763	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → 𝑚 = 𝑚)
28	22, 26, 27	oveq123d 7438	. . . . . . . . 9 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (((𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)‘𝑘)(⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩𝑂((1st ‘⟨𝐹, 𝐺⟩)‘𝑦))𝑚) = (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚))
29	28	eqeq2d 2773	. . . . . . . 8 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑔 = (((𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)‘𝑘)(⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩𝑂((1st ‘⟨𝐹, 𝐺⟩)‘𝑦))𝑚) ↔ 𝑔 = (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚)))
30	29	reubidv 3383	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)‘𝑘)(⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩𝑂((1st ‘⟨𝐹, 𝐺⟩)‘𝑦))𝑚) ↔ ∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚)))
31	20, 30	raleqbidv 3336	. . . . . 6 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (∀𝑔 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)‘𝑘)(⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩𝑂((1st ‘⟨𝐹, 𝐺⟩)‘𝑦))𝑚) ↔ ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚)))
32	31	ralbidv 3187	. . . . 5 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)‘𝑘)(⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩𝑂((1st ‘⟨𝐹, 𝐺⟩)‘𝑦))𝑚) ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚)))
33	18, 32	anbi12d 644	. . . 4 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)‘𝑘)(⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩𝑂((1st ‘⟨𝐹, 𝐺⟩)‘𝑦))𝑚)) ↔ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽(𝐹‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚))))
34	33	opabbidv 5175	. . 3 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)‘𝑘)(⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩𝑂((1st ‘⟨𝐹, 𝐺⟩)‘𝑦))𝑚))} = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽(𝐹‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚))})
35	7, 34	syl 18	. 2 ⊢ (𝜑 → {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)‘𝑘)(⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩𝑂((1st ‘⟨𝐹, 𝐺⟩)‘𝑦))𝑚))} = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽(𝐹‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚))})
36	10, 35	eqtrd 2797	1 ⊢ (𝜑 → (⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊) = {⟨𝑥, 𝑚⟩ ∣ ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽(𝐹‘𝑥))) ∧ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚))})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃!wreu 3365  Vcvv 3453  ⟨cop 4593   class class class wbr 5107  {copab 5171  ‘cfv 6537  (class class class)co 7417  1^st c1st 7988  2^nd c2nd 7989  Basecbs 17307  Hom chom 17359  compcco 17360   Func cfunc 17949   UP cup 50107

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7740

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7420  df-oprab 7421  df-mpo 7422  df-1st 7990  df-2nd 7991  df-func 17953  df-up 50108

This theorem is used by:  isuplem  50113