Theorem upfval3 49086

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 5116	. . . 4 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
9	7, 8	sylib 218	. . 3 ⊢ (𝜑 → ⟨𝐹, 𝐺⟩ ∈ (𝐷 Func 𝐸))
10	1, 2, 3, 4, 5, 6, 9	upfval2 49085	. 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 17830	. . . . . . . . . . 11 ⊢ Rel (𝐷 Func 𝐸)
12	11	brrelex12i 5701	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
13		op1stg 7989	. . . . . . . . . 10 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
14	12, 13	syl 17	. . . . . . . . 9 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (1st ‘⟨𝐹, 𝐺⟩) = 𝐹)
15	14	fveq1d 6867	. . . . . . . 8 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥) = (𝐹‘𝑥))
16	15	oveq2d 7410	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)) = (𝑊𝐽(𝐹‘𝑥)))
17	16	eleq2d 2815	. . . . . 6 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑚 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)) ↔ 𝑚 ∈ (𝑊𝐽(𝐹‘𝑥))))
18	17	anbi2d 630	. . . . 5 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑥))) ↔ (𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽(𝐹‘𝑥)))))
19	14	fveq1d 6867	. . . . . . . 8 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑦) = (𝐹‘𝑦))
20	19	oveq2d 7410	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = (𝑊𝐽(𝐹‘𝑦)))
21	15	opeq2d 4852	. . . . . . . . . . 11 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩ = ⟨𝑊, (𝐹‘𝑥)⟩)
22	21, 19	oveq12d 7412	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩𝑂((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = (⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦)))
23		op2ndg 7990	. . . . . . . . . . . . 13 ⊢ ((𝐹 ∈ V ∧ 𝐺 ∈ V) → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
24	12, 23	syl 17	. . . . . . . . . . . 12 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (2nd ‘⟨𝐹, 𝐺⟩) = 𝐺)
25	24	oveqd 7411	. . . . . . . . . . 11 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦) = (𝑥𝐺𝑦))
26	25	fveq1d 6867	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ((𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)‘𝑘) = ((𝑥𝐺𝑦)‘𝑘))
27		eqidd 2731	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → 𝑚 = 𝑚)
28	22, 26, 27	oveq123d 7415	. . . . . . . . 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 2741	. . . . . . . 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 3375	. . . . . . 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 3322	. . . . . 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 3158	. . . . 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 632	. . . 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 5181	. . 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 17	. 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 2765	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))})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3046  ∃!wreu 3355  Vcvv 3455  ⟨cop 4603   class class class wbr 5115  {copab 5177  ‘cfv 6519  (class class class)co 7394  1^st c1st 7975  2^nd c2nd 7976  Basecbs 17185  Hom chom 17237  compcco 17238   Func cfunc 17822   UP cup 49081

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 2702  ax-rep 5242  ax-sep 5259  ax-nul 5269  ax-pow 5328  ax-pr 5395  ax-un 7718

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2880  df-ne 2928  df-ral 3047  df-rex 3056  df-reu 3358  df-rab 3412  df-v 3457  df-sbc 3762  df-csb 3871  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-iun 4965  df-br 5116  df-opab 5178  df-mpt 5197  df-id 5541  df-xp 5652  df-rel 5653  df-cnv 5654  df-co 5655  df-dm 5656  df-rn 5657  df-res 5658  df-ima 5659  df-iota 6472  df-fun 6521  df-fn 6522  df-f 6523  df-f1 6524  df-fo 6525  df-f1o 6526  df-fv 6527  df-ov 7397  df-oprab 7398  df-mpo 7399  df-1st 7977  df-2nd 7978  df-func 17826  df-up 49082

This theorem is referenced by:  isuplem  49087