Theorem isup2 49101

Description: The universal property of a universal pair. (Contributed by Zhi Wang, 24-Sep-2025.)

Hypotheses

Ref	Expression
isup2.b	⊢ 𝐵 = (Base‘𝐷)
isup2.h	⊢ 𝐻 = (Hom ‘𝐷)
isup2.j	⊢ 𝐽 = (Hom ‘𝐸)
isup2.o	⊢ 𝑂 = (comp‘𝐸)
isup2.x	⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)

Assertion

Ref	Expression
isup2	⊢ (𝜑 → ∀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   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 isup2

Step	Hyp	Ref	Expression
1		isup2.x	. 2 ⊢ (𝜑 → 𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀)
2		isup2.b	. . 3 ⊢ 𝐵 = (Base‘𝐷)
3		eqid 2730	. . 3 ⊢ (Base‘𝐸) = (Base‘𝐸)
4		isup2.h	. . 3 ⊢ 𝐻 = (Hom ‘𝐷)
5		isup2.j	. . 3 ⊢ 𝐽 = (Hom ‘𝐸)
6		isup2.o	. . 3 ⊢ 𝑂 = (comp‘𝐸)
7	1, 3	uprcl3 49097	. . 3 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
8	1	uprcl2 49096	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
9	1, 2	uprcl4 49098	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10	1, 5	uprcl5 49099	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋)))
11	2, 3, 4, 5, 6, 7, 8, 9, 10	isup 49088	. 2 ⊢ (𝜑 → (𝑋(⟨𝐹, 𝐺⟩(𝐷 UP 𝐸)𝑊)𝑀 ↔ ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀)))
12	1, 11	mpbid 232	1 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀))

Syntax hints:   → wi 4   = wceq 1540  ∀wral 3046  ∃!wreu 3355  ⟨cop 4603   class class class wbr 5115  ‘cfv 6519  (class class class)co 7394  Basecbs 17185  Hom chom 17237  compcco 17238   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:  upeu3  49102  upeu4  49103