Theorem iinfssclem2

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
Assertion	iinfssclem2	⊢ ( 𝜑 → 𝐾 Fn ( ∩ 𝑥 ∈ 𝐴 𝑆 × ∩ 𝑥 ∈ 𝐴 𝑆 ) )

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		ovex	⊢ ( 𝑧 𝐻 𝑤 ) ∈ V
7	6	rgenw	⊢ ∀ 𝑥 ∈ 𝐴 ( 𝑧 𝐻 𝑤 ) ∈ V
8		iinexg	⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐴 ( 𝑧 𝐻 𝑤 ) ∈ V ) → ∩ 𝑥 ∈ 𝐴 ( 𝑧 𝐻 𝑤 ) ∈ V )
9	1 7 8	sylancl	⊢ ( 𝜑 → ∩ 𝑥 ∈ 𝐴 ( 𝑧 𝐻 𝑤 ) ∈ V )
10	9	adantr	⊢ ( ( 𝜑 ∧ ( 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ∧ 𝑤 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ) ) → ∩ 𝑥 ∈ 𝐴 ( 𝑧 𝐻 𝑤 ) ∈ V )
11	10	ralrimivva	⊢ ( 𝜑 → ∀ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ∀ 𝑤 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ∩ 𝑥 ∈ 𝐴 ( 𝑧 𝐻 𝑤 ) ∈ V )
12		eqid	⊢ ( 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 , 𝑤 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ↦ ∩ 𝑥 ∈ 𝐴 ( 𝑧 𝐻 𝑤 ) ) = ( 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 , 𝑤 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ↦ ∩ 𝑥 ∈ 𝐴 ( 𝑧 𝐻 𝑤 ) )
13	12	fnmpo	⊢ ( ∀ 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ∀ 𝑤 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ∩ 𝑥 ∈ 𝐴 ( 𝑧 𝐻 𝑤 ) ∈ V → ( 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 , 𝑤 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ↦ ∩ 𝑥 ∈ 𝐴 ( 𝑧 𝐻 𝑤 ) ) Fn ( ∩ 𝑥 ∈ 𝐴 𝑆 × ∩ 𝑥 ∈ 𝐴 𝑆 ) )
14	11 13	syl	⊢ ( 𝜑 → ( 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 , 𝑤 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ↦ ∩ 𝑥 ∈ 𝐴 ( 𝑧 𝐻 𝑤 ) ) Fn ( ∩ 𝑥 ∈ 𝐴 𝑆 × ∩ 𝑥 ∈ 𝐴 𝑆 ) )
15	1 2 3 4 5	iinfssclem1	⊢ ( 𝜑 → 𝐾 = ( 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 , 𝑤 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ↦ ∩ 𝑥 ∈ 𝐴 ( 𝑧 𝐻 𝑤 ) ) )
16	15	fneq1d	⊢ ( 𝜑 → ( 𝐾 Fn ( ∩ 𝑥 ∈ 𝐴 𝑆 × ∩ 𝑥 ∈ 𝐴 𝑆 ) ↔ ( 𝑧 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 , 𝑤 ∈ ∩ 𝑥 ∈ 𝐴 𝑆 ↦ ∩ 𝑥 ∈ 𝐴 ( 𝑧 𝐻 𝑤 ) ) Fn ( ∩ 𝑥 ∈ 𝐴 𝑆 × ∩ 𝑥 ∈ 𝐴 𝑆 ) ) )
17	14 16	mpbird	⊢ ( 𝜑 → 𝐾 Fn ( ∩ 𝑥 ∈ 𝐴 𝑆 × ∩ 𝑥 ∈ 𝐴 𝑆 ) )