Theorem f1owe

Description: Well-ordering of isomorphic relations. (Contributed by NM, 4-Mar-1997)

	Ref	Expression
Hypothesis	f1owe.1	⊢ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝐹 ‘ 𝑥 ) 𝑆 ( 𝐹 ‘ 𝑦 ) }
Assertion	f1owe	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ( 𝑆 We 𝐵 → 𝑅 We 𝐴 ) )

Proof

Step	Hyp	Ref	Expression
1		f1owe.1	⊢ 𝑅 = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝐹 ‘ 𝑥 ) 𝑆 ( 𝐹 ‘ 𝑦 ) }
2		fveq2	⊢ ( 𝑥 = 𝑧 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑧 ) )
3	2	breq1d	⊢ ( 𝑥 = 𝑧 → ( ( 𝐹 ‘ 𝑥 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑧 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ) )
4		fveq2	⊢ ( 𝑦 = 𝑤 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑤 ) )
5	4	breq2d	⊢ ( 𝑦 = 𝑤 → ( ( 𝐹 ‘ 𝑧 ) 𝑆 ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑧 ) 𝑆 ( 𝐹 ‘ 𝑤 ) ) )
6	3 5 1	brabg	⊢ ( ( 𝑧 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴 ) → ( 𝑧 𝑅 𝑤 ↔ ( 𝐹 ‘ 𝑧 ) 𝑆 ( 𝐹 ‘ 𝑤 ) ) )
7	6	rgen2	⊢ ∀ 𝑧 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( 𝑧 𝑅 𝑤 ↔ ( 𝐹 ‘ 𝑧 ) 𝑆 ( 𝐹 ‘ 𝑤 ) )
8		df-isom	⊢ ( 𝐹 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) ↔ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ ∀ 𝑧 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( 𝑧 𝑅 𝑤 ↔ ( 𝐹 ‘ 𝑧 ) 𝑆 ( 𝐹 ‘ 𝑤 ) ) ) )
9		isowe	⊢ ( 𝐹 Isom 𝑅 , 𝑆 ( 𝐴 , 𝐵 ) → ( 𝑅 We 𝐴 ↔ 𝑆 We 𝐵 ) )
10	8 9	sylbir	⊢ ( ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 ∧ ∀ 𝑧 ∈ 𝐴 ∀ 𝑤 ∈ 𝐴 ( 𝑧 𝑅 𝑤 ↔ ( 𝐹 ‘ 𝑧 ) 𝑆 ( 𝐹 ‘ 𝑤 ) ) ) → ( 𝑅 We 𝐴 ↔ 𝑆 We 𝐵 ) )
11	7 10	mpan2	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ( 𝑅 We 𝐴 ↔ 𝑆 We 𝐵 ) )
12	11	biimprd	⊢ ( 𝐹 : 𝐴 –1-1-onto→ 𝐵 → ( 𝑆 We 𝐵 → 𝑅 We 𝐴 ) )