Theorem upfval3 50388

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 5103	. . . 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 50387	. 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 18043	. . . . . . . . . . 11 ⊢ Rel (𝐷 Func 𝐸)
12	11	brrelex12i 5705	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
13		op1stg 8014	. . . . . . . . . 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 6888	. . . . . . . 8 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥) = (𝐹‘𝑥))
16	15	oveq2d 7437	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)) = (𝑊𝐽(𝐹‘𝑥)))
17	16	eleq2d 2846	. . . . . 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 6888	. . . . . . . 8 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑦) = (𝐹‘𝑦))
20	19	oveq2d 7437	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = (𝑊𝐽(𝐹‘𝑦)))
21	15	opeq2d 4846	. . . . . . . . . . 11 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩ = ⟨𝑊, (𝐹‘𝑥)⟩)
22	21, 19	oveq12d 7439	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩𝑂((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = (⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦)))
23		op2ndg 8015	. . . . . . . . . . . . 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 7438	. . . . . . . . . . 11 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦) = (𝑥𝐺𝑦))
26	25	fveq1d 6888	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ((𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)‘𝑘) = ((𝑥𝐺𝑦)‘𝑘))
27		eqidd 2761	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → 𝑚 = 𝑚)
28	22, 26, 27	oveq123d 7442	. . . . . . . . 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 2771	. . . . . . . 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 3381	. . . . . . 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 3334	. . . . . 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 3185	. . . . 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 5170	. . 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 2795	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 1569   ∈ wcel 2144  ∀wral 3076  ∃!wreu 3363  Vcvv 3450  ⟨cop 4834   class class class wbr 5102  {copab 5166  ‘cfv 6539  (class class class)co 7421  1^st c1st 8000  2^nd c2nd 8001  Basecbs 17393  Hom chom 17445  compcco 17446   Func cfunc 18035   UP cup 50383

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-func 18039  df-up 50384

This theorem is used by:  isuplem  50389