Theorem isowe2

Description: A weak form of isowe that does not need Replacement. (Contributed by Mario Carneiro, 18-Nov-2014)

	Ref	Expression
Assertion	isowe2	⊢ ( ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ∧ ∀ 𝑥 ( 𝐻 “ 𝑥 ) ∈ V ) → ( 𝑆 We 𝐵 → 𝑅 We 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		simpl	⊢ ( ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ∧ ∀ 𝑥 ( 𝐻 “ 𝑥 ) ∈ V ) → 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) )
2		imaeq2	⊢ ( 𝑥 = 𝑦 → ( 𝐻 “ 𝑥 ) = ( 𝐻 “ 𝑦 ) )
3	2	eleq1d	⊢ ( 𝑥 = 𝑦 → ( ( 𝐻 “ 𝑥 ) ∈ V ↔ ( 𝐻 “ 𝑦 ) ∈ V ) )
4	3	spvv	⊢ ( ∀ 𝑥 ( 𝐻 “ 𝑥 ) ∈ V → ( 𝐻 “ 𝑦 ) ∈ V )
5	4	adantl	⊢ ( ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ∧ ∀ 𝑥 ( 𝐻 “ 𝑥 ) ∈ V ) → ( 𝐻 “ 𝑦 ) ∈ V )
6	1 5	isofrlem	⊢ ( ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ∧ ∀ 𝑥 ( 𝐻 “ 𝑥 ) ∈ V ) → ( 𝑆 Fr 𝐵 → 𝑅 Fr 𝐴 ) )
7		isosolem	⊢ ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑆 Or 𝐵 → 𝑅 Or 𝐴 ) )
8	7	adantr	⊢ ( ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ∧ ∀ 𝑥 ( 𝐻 “ 𝑥 ) ∈ V ) → ( 𝑆 Or 𝐵 → 𝑅 Or 𝐴 ) )
9	6 8	anim12d	⊢ ( ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ∧ ∀ 𝑥 ( 𝐻 “ 𝑥 ) ∈ V ) → ( ( 𝑆 Fr 𝐵 ∧ 𝑆 Or 𝐵 ) → ( 𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴 ) ) )
10		df-we	⊢ ( 𝑆 We 𝐵 ↔ ( 𝑆 Fr 𝐵 ∧ 𝑆 Or 𝐵 ) )
11		df-we	⊢ ( 𝑅 We 𝐴 ↔ ( 𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴 ) )
12	9 10 11	3imtr4g	⊢ ( ( 𝐻 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ∧ ∀ 𝑥 ( 𝐻 “ 𝑥 ) ∈ V ) → ( 𝑆 We 𝐵 → 𝑅 We 𝐴 ) )