Theorem uptposlem

Description: Lemma for uptpos . (Contributed by Zhi Wang, 4-Nov-2025)

	Ref	Expression
Hypotheses	oppcuprcl2.x	⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝑂 UP 𝑃 ) 𝑊 ) 𝑀 )
	uptpos.h	⊢ ( 𝜑 → tpos 𝐺 = 𝐻 )
Assertion	uptposlem	⊢ ( 𝜑 → tpos 𝐻 = 𝐺 )

Proof

Step	Hyp	Ref	Expression
1		oppcuprcl2.x	⊢ ( 𝜑 → 𝑋 ( ⟨ 𝐹 , 𝐺 ⟩ ( 𝑂 UP 𝑃 ) 𝑊 ) 𝑀 )
2		uptpos.h	⊢ ( 𝜑 → tpos 𝐺 = 𝐻 )
3	2	tposeqd	⊢ ( 𝜑 → tpos tpos 𝐺 = tpos 𝐻 )
4		eqid	⊢ ( Base ‘ 𝑂 ) = ( Base ‘ 𝑂 )
5	1	uprcl2	⊢ ( 𝜑 → 𝐹 ( 𝑂 Func 𝑃 ) 𝐺 )
6	4 5	funcfn2	⊢ ( 𝜑 → 𝐺 Fn ( ( Base ‘ 𝑂 ) × ( Base ‘ 𝑂 ) ) )
7		fnrel	⊢ ( 𝐺 Fn ( ( Base ‘ 𝑂 ) × ( Base ‘ 𝑂 ) ) → Rel 𝐺 )
8	6 7	syl	⊢ ( 𝜑 → Rel 𝐺 )
9		relxp	⊢ Rel ( ( Base ‘ 𝑂 ) × ( Base ‘ 𝑂 ) )
10	6	fndmd	⊢ ( 𝜑 → dom 𝐺 = ( ( Base ‘ 𝑂 ) × ( Base ‘ 𝑂 ) ) )
11	10	releqd	⊢ ( 𝜑 → ( Rel dom 𝐺 ↔ Rel ( ( Base ‘ 𝑂 ) × ( Base ‘ 𝑂 ) ) ) )
12	9 11	mpbiri	⊢ ( 𝜑 → Rel dom 𝐺 )
13		tpostpos2	⊢ ( ( Rel 𝐺 ∧ Rel dom 𝐺 ) → tpos tpos 𝐺 = 𝐺 )
14	8 12 13	syl2anc	⊢ ( 𝜑 → tpos tpos 𝐺 = 𝐺 )
15	3 14	eqtr3d	⊢ ( 𝜑 → tpos 𝐻 = 𝐺 )