Theorem isuplem 49808

Description: Lemma for isup 49809 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 49807	. 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 7407	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥𝐻𝑦) = (𝑋𝐻𝑦))
10		fveq2 6871	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
11	10	opeq2d 4841	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ⟨𝑊, (𝐹‘𝑥)⟩ = ⟨𝑊, (𝐹‘𝑋)⟩)
12	11	oveq1d 7415	. . . . . 6 ⊢ (𝑥 = 𝑋 → (⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦)) = (⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦)))
13		oveq1 7407	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥𝐺𝑦) = (𝑋𝐺𝑦))
14	13	fveq1d 6873	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥𝐺𝑦)‘𝑘) = ((𝑋𝐺𝑦)‘𝑘))
15		eqidd 2766	. . . . . 6 ⊢ (𝑥 = 𝑋 → 𝑚 = 𝑚)
16	12, 14, 15	oveq123d 7421	. . . . 5 ⊢ (𝑥 = 𝑋 → (((𝑥𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑥)⟩𝑂(𝐹‘𝑦))𝑚) = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑚))
17	16	eqeq2d 2776	. . . 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 3406	. . 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 3229	. 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 7408	. . . . 5 ⊢ (𝑚 = 𝑀 → (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑚) = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀))
21	20	eqeq2d 2776	. . . 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 3386	. . 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 3229	. 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 2766	. 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑚 = 𝑀) → 𝐵 = 𝐵)
25		simpl 487	. . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑚 = 𝑀) → 𝑥 = 𝑋)
26	25	fveq2d 6875	. . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑚 = 𝑀) → (𝐹‘𝑥) = (𝐹‘𝑋))
27	26	oveq2d 7416	. 2 ⊢ ((𝑥 = 𝑋 ∧ 𝑚 = 𝑀) → (𝑊𝐽(𝐹‘𝑥)) = (𝑊𝐽(𝐹‘𝑋)))
28	8, 19, 23, 24, 27	brab2ddw 49458	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 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  ∃!wreu 3368  ⟨cop 4591   class class class wbr 5105  ‘cfv 6525  (class class class)co 7400  Basecbs 17259  Hom chom 17311  compcco 17312   Func cfunc 17901   UP cup 49802

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-iun 4954  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-1st 7974  df-2nd 7975  df-func 17905  df-up 49803

This theorem is referenced by:  isup  49809  uprcl4  49820  uprcl5  49821