Theorem iinfssclem3

Description: Lemma for iinfssc . (Contributed by Zhi Wang, 31-Oct-2025)

	Ref	Expression
Hypotheses	iinfssc.1	⊢ ( 𝜑 → 𝐴 ≠ ∅ )
	iinfssc.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐻 ⊆cat 𝐽 )
	iinfssc.3	⊢ ( 𝜑 → 𝐾 = ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 ( 𝐻 ‘ 𝑦 ) ) )
	iinfssclem1.4	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑆 = dom dom 𝐻 )
	iinfssclem1.5	⊢ Ⅎ 𝑥 𝜑
	iinfssclem3.x	⊢ ( 𝜑 → 𝑋 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 )
	iinfssclem3.y	⊢ ( 𝜑 → 𝑌 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 )
Assertion	iinfssclem3	⊢ ( 𝜑 → ( 𝑋 𝐾 𝑌 ) = ∩ 𝑥 ∈ 𝐴 ( 𝑋 𝐻 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		iinfssc.1	⊢ ( 𝜑 → 𝐴 ≠ ∅ )
2		iinfssc.2	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐻 ⊆cat 𝐽 )
3		iinfssc.3	⊢ ( 𝜑 → 𝐾 = ( 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 dom 𝐻 ↦ ∩ 𝑥 ∈ 𝐴 ( 𝐻 ‘ 𝑦 ) ) )
4		iinfssclem1.4	⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑆 = dom dom 𝐻 )
5		iinfssclem1.5	⊢ Ⅎ 𝑥 𝜑
6		iinfssclem3.x	⊢ ( 𝜑 → 𝑋 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 )
7		iinfssclem3.y	⊢ ( 𝜑 → 𝑌 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 )
8	1 2 3 4 5	iinfssclem1	⊢ ( 𝜑 → 𝐾 = ( 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 , 𝑤 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ↦ ∩ 𝑥 ∈ 𝐴 ( 𝑧 𝐻 𝑤 ) ) )
9		nfv	⊢ Ⅎ 𝑥 ( 𝑧 = 𝑋 ∧ 𝑤 = 𝑌 )
10	5 9	nfan	⊢ Ⅎ 𝑥 ( 𝜑 ∧ ( 𝑧 = 𝑋 ∧ 𝑤 = 𝑌 ) )
11		simplrl	⊢ ( ( ( 𝜑 ∧ ( 𝑧 = 𝑋 ∧ 𝑤 = 𝑌 ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝑧 = 𝑋 )
12		simplrr	⊢ ( ( ( 𝜑 ∧ ( 𝑧 = 𝑋 ∧ 𝑤 = 𝑌 ) ) ∧ 𝑥 ∈ 𝐴 ) → 𝑤 = 𝑌 )
13	11 12	oveq12d	⊢ ( ( ( 𝜑 ∧ ( 𝑧 = 𝑋 ∧ 𝑤 = 𝑌 ) ) ∧ 𝑥 ∈ 𝐴 ) → ( 𝑧 𝐻 𝑤 ) = ( 𝑋 𝐻 𝑌 ) )
14	10 13	iineq2d	⊢ ( ( 𝜑 ∧ ( 𝑧 = 𝑋 ∧ 𝑤 = 𝑌 ) ) → ∩ 𝑥 ∈ 𝐴 ( 𝑧 𝐻 𝑤 ) = ∩ 𝑥 ∈ 𝐴 ( 𝑋 𝐻 𝑌 ) )
15		ovex	⊢ ( 𝑋 𝐻 𝑌 ) ∈ V
16	15	rgenw	⊢ ∀ 𝑥 ∈ 𝐴 ( 𝑋 𝐻 𝑌 ) ∈ V
17		iinexg	⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑋 𝐻 𝑌 ) ∈ V ) → ∩ 𝑥 ∈ 𝐴 ( 𝑋 𝐻 𝑌 ) ∈ V )
18	1 16 17	sylancl	⊢ ( 𝜑 → ∩ 𝑥 ∈ 𝐴 ( 𝑋 𝐻 𝑌 ) ∈ V )
19	8 14 6 7 18	ovmpod	⊢ ( 𝜑 → ( 𝑋 𝐾 𝑌 ) = ∩ 𝑥 ∈ 𝐴 ( 𝑋 𝐻 𝑌 ) )