Theorem isup2 50403

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 2760	. . 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 50399	. . 3 ⊢ (𝜑 → 𝑊 ∈ (Base‘𝐸))
8	1	uprcl2 50398	. . 3 ⊢ (𝜑 → 𝐹(𝐷 Func 𝐸)𝐺)
9	1, 2	uprcl4 50400	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
10	1, 5	uprcl5 50401	. . 3 ⊢ (𝜑 → 𝑀 ∈ (𝑊𝐽(𝐹‘𝑋)))
11	2, 3, 4, 5, 6, 7, 8, 9, 10	isup 50389	. 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 235	1 ⊢ (𝜑 → ∀𝑦 ∈ 𝐵 ∀𝑔 ∈ (𝑊𝐽(𝐹‘𝑦))∃!𝑘 ∈ (𝑋𝐻𝑦)𝑔 = (((𝑋𝐺𝑦)‘𝑘)(⟨𝑊, (𝐹‘𝑋)⟩𝑂(𝐹‘𝑦))𝑀))

This proof depends on syntax axioms:   → wi 4   = wceq 1569  ∀wral 3076  ∃!wreu 3363  ⟨cop 4834   class class class wbr 5102  ‘cfv 6538  (class class class)co 7420  Basecbs 17392  Hom chom 17444  compcco 17445   UP cup 50382

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-func 18038  df-up 50383

This theorem is used by:  upeu3  50404  upeu4  50405