Theorem ssclem

Description: Lemma for ssc1 and similar theorems. (Contributed by Mario Carneiro, 6-Jan-2017)

	Ref	Expression
Hypothesis	isssc.1	⊢ ( 𝜑 → 𝐻 Fn ( 𝑆 × 𝑆 ) )
Assertion	ssclem	⊢ ( 𝜑 → ( 𝐻 ∈ V ↔ 𝑆 ∈ V ) )

Proof

Step	Hyp	Ref	Expression
1		isssc.1	⊢ ( 𝜑 → 𝐻 Fn ( 𝑆 × 𝑆 ) )
2		dmxpid	⊢ dom ( 𝑆 × 𝑆 ) = 𝑆
3	1	fndmd	⊢ ( 𝜑 → dom 𝐻 = ( 𝑆 × 𝑆 ) )
4	3	adantr	⊢ ( ( 𝜑 ∧ 𝐻 ∈ V ) → dom 𝐻 = ( 𝑆 × 𝑆 ) )
5		dmexg	⊢ ( 𝐻 ∈ V → dom 𝐻 ∈ V )
6	5	adantl	⊢ ( ( 𝜑 ∧ 𝐻 ∈ V ) → dom 𝐻 ∈ V )
7	4 6	eqeltrrd	⊢ ( ( 𝜑 ∧ 𝐻 ∈ V ) → ( 𝑆 × 𝑆 ) ∈ V )
8	7	dmexd	⊢ ( ( 𝜑 ∧ 𝐻 ∈ V ) → dom ( 𝑆 × 𝑆 ) ∈ V )
9	2 8	eqeltrrid	⊢ ( ( 𝜑 ∧ 𝐻 ∈ V ) → 𝑆 ∈ V )
10		sqxpexg	⊢ ( 𝑆 ∈ V → ( 𝑆 × 𝑆 ) ∈ V )
11		fnex	⊢ ( ( 𝐻 Fn ( 𝑆 × 𝑆 ) ∧ ( 𝑆 × 𝑆 ) ∈ V ) → 𝐻 ∈ V )
12	1 10 11	syl2an	⊢ ( ( 𝜑 ∧ 𝑆 ∈ V ) → 𝐻 ∈ V )
13	9 12	impbida	⊢ ( 𝜑 → ( 𝐻 ∈ V ↔ 𝑆 ∈ V ) )