Theorem isuplem 49666

Description: Lemma for isup 49667 and other theorems. (Contributed by Zhi Wang, 25-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
isuplem	⊢ (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)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,𝑦   𝐶,𝑔,𝑘,𝑦   𝐷,𝑔,𝑘,𝑦   𝑔,𝐸,𝑘,𝑦   𝑔,𝐹,𝑘,𝑦   𝑔,𝐺,𝑘,𝑦   𝑔,𝐻,𝑘,𝑦   𝑔,𝐽,𝑘,𝑦   𝑔,𝑀,𝑘,𝑦   𝑔,𝑂,𝑘,𝑦   𝑔,𝑊,𝑘,𝑦   𝑔,𝑋,𝑘,𝑦

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

Proof of Theorem isuplem

Dummy variables 𝑚 𝑥 are mutually distinct and distinct from all other variables.

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	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
8	1, 2, 3, 4, 5, 6, 7	upfval3 49665	. 2 ⊢ (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))})
9		oveq1 7367	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
10		fveq2 6834	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
11	10	opeq2d 4824	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ⟨𝑊, (𝐹‘𝑥)⟩ = ⟨𝑊, (𝐹‘𝑋)⟩)
12	11	oveq1d 7375	. . . . . 6 ⊢ (𝑥 = 𝑋 → (⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦)) = (⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦)))
13		oveq1 7367	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥𝐺𝑦) = (𝑋𝐺𝑦))
14	13	fveq1d 6836	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥𝐺𝑦)‘𝑘) = ((𝑋𝐺𝑦)‘𝑘))
15		eqidd 2738	. . . . . 6 ⊢ (𝑥 = 𝑋 → 𝑚 = 𝑚)
16	12, 14, 15	oveq123d 7381	. . . . 5 ⊢ (𝑥 = 𝑋 → (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚) = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑚))
17	16	eqeq2d 2748	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑔 = (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚) ↔ 𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑚)))
18	9, 17	reueqbidv 3379	. . 3 ⊢ (𝑥 = 𝑋 → (∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚) ↔ ∃!𝑘 ∈ (𝑋𝐻𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑚)))
19	18	2ralbidv 3202	. 2 ⊢ (𝑥 = 𝑋 → (∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑥𝐻𝑦)𝑔 = (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚) ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑚)))
20		oveq2 7368	. . . . 5 ⊢ (𝑚 = 𝑀 → (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑚) = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀))
21	20	eqeq2d 2748	. . . 4 ⊢ (𝑚 = 𝑀 → (𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑚) ↔ 𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀)))
22	21	reubidv 3359	. . 3 ⊢ (𝑚 = 𝑀 → (∃!𝑘 ∈ (𝑋𝐻𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑚) ↔ ∃!𝑘 ∈ (𝑋𝐻𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀)))
23	22	2ralbidv 3202	. 2 ⊢ (𝑚 = 𝑀 → (∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑚) ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀)))
24		eqidd 2738	. 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑚 = 𝑀) → 𝐵 = 𝐵)
25		simpl 482	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑚 = 𝑀) → 𝑥 = 𝑋)
26	25	fveq2d 6838	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑚 = 𝑀) → (𝐹‘𝑥) = (𝐹‘𝑋))
27	26	oveq2d 7376	. 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑚 = 𝑀) → (𝑊𝐽(𝐹‘𝑥)) = (𝑊𝐽(𝐹‘𝑋)))
28	8, 19, 23, 24, 27	brab2ddw 49316	1 ⊢ (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)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   ↔ wb 206   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃!wreu 3341  ⟨cop 4574   class class class wbr 5086  ‘cfv 6492  (class class class)co 7360  Basecbs 17170  Hom chom 17222  compcco 17223   Func cfunc 17812   UP cup 49660

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 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

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 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-func 17816  df-up 49661

This theorem is referenced by:  isup  49667  uprcl4  49678  uprcl5  49679