Theorem upfval3 49531

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 5101	. . . 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 49530	. 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 17798	. . . . . . . . . . 11 ⊢ Rel (𝐷 Func 𝐸)
12	11	brrelex12i 5687	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝐹 ∈ V ∧ 𝐺 ∈ V))
13		op1stg 7955	. . . . . . . . . 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 6844	. . . . . . . 8 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥) = (𝐹‘𝑥))
16	15	oveq2d 7384	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)) = (𝑊𝐽(𝐹‘𝑥)))
17	16	eleq2d 2823	. . . . . 6 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑚 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)) ↔ 𝑚 ∈ (𝑊𝐽(𝐹‘𝑥))))
18	17	anbi2d 631	. . . . 5 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ((𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑥))) ↔ (𝑥 ∈ 𝐵 ∧ 𝑚 ∈ (𝑊𝐽(𝐹‘𝑥)))))
19	14	fveq1d 6844	. . . . . . . 8 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ((1st ‘⟨𝐹, 𝐺⟩)‘𝑦) = (𝐹‘𝑦))
20	19	oveq2d 7384	. . . . . . 7 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑊𝐽((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = (𝑊𝐽(𝐹‘𝑦)))
21	15	opeq2d 4838	. . . . . . . . . . 11 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩ = ⟨𝑊, (𝐹‘𝑥)⟩)
22	21, 19	oveq12d 7386	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (⟨𝑊, ((1st ‘⟨𝐹, 𝐺⟩)‘𝑥)⟩𝑂((1st ‘⟨𝐹, 𝐺⟩)‘𝑦)) = (⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦)))
23		op2ndg 7956	. . . . . . . . . . . . 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 7385	. . . . . . . . . . 11 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → (𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦) = (𝑥𝐺𝑦))
26	25	fveq1d 6844	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → ((𝑥(2nd ‘⟨𝐹, 𝐺⟩)𝑦)‘𝑘) = ((𝑥𝐺𝑦)‘𝑘))
27		eqidd 2738	. . . . . . . . . 10 ⊢ (𝐹(𝐷 Func 𝐸)𝐺 → 𝑚 = 𝑚)
28	22, 26, 27	oveq123d 7389	. . . . . . . . 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 2748	. . . . . . . 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 3368	. . . . . . 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 3318	. . . . . 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 3161	. . . . 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 633	. . . 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 5166	. . 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 2772	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 1542   ∈ wcel 2114  ∀wral 3052  ∃!wreu 3350  Vcvv 3442  ⟨cop 4588   class class class wbr 5100  {copab 5162  ‘cfv 6500  (class class class)co 7368  1^st c1st 7941  2^nd c2nd 7942  Basecbs 17148  Hom chom 17200  compcco 17201   Func cfunc 17790   UP cup 49526

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-func 17794  df-up 49527

This theorem is referenced by:  isuplem  49532