Theorem ispautN

Description: The predictate "is a projective automorphism." (Contributed by NM, 26-Jan-2012) (New usage is discouraged.)

	Ref	Expression
Hypotheses	pautset.s	⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
	pautset.m	⊢ 𝑀 = ( PAut ‘ 𝐾 )
Assertion	ispautN	⊢ ( 𝐾 ∈ 𝐵 → ( 𝐹 ∈ 𝑀 ↔ ( 𝐹 : 𝑆 –1-1-onto→ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ⊆ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ⊆ ( 𝐹 ‘ 𝑦 ) ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		pautset.s	⊢ 𝑆 = ( PSubSp ‘ 𝐾 )
2		pautset.m	⊢ 𝑀 = ( PAut ‘ 𝐾 )
3	1 2	pautsetN	⊢ ( 𝐾 ∈ 𝐵 → 𝑀 = { 𝑓 ∣ ( 𝑓 : 𝑆 –1-1-onto→ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ⊆ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ⊆ ( 𝑓 ‘ 𝑦 ) ) ) } )
4	3	eleq2d	⊢ ( 𝐾 ∈ 𝐵 → ( 𝐹 ∈ 𝑀 ↔ 𝐹 ∈ { 𝑓 ∣ ( 𝑓 : 𝑆 –1-1-onto→ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ⊆ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ⊆ ( 𝑓 ‘ 𝑦 ) ) ) } ) )
5		f1of	⊢ ( 𝐹 : 𝑆 –1-1-onto→ 𝑆 → 𝐹 : 𝑆 ⟶ 𝑆 )
6	1	fvexi	⊢ 𝑆 ∈ V
7		fex	⊢ ( ( 𝐹 : 𝑆 ⟶ 𝑆 ∧ 𝑆 ∈ V ) → 𝐹 ∈ V )
8	5 6 7	sylancl	⊢ ( 𝐹 : 𝑆 –1-1-onto→ 𝑆 → 𝐹 ∈ V )
9	8	adantr	⊢ ( ( 𝐹 : 𝑆 –1-1-onto→ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ⊆ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ⊆ ( 𝐹 ‘ 𝑦 ) ) ) → 𝐹 ∈ V )
10		f1oeq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 : 𝑆 –1-1-onto→ 𝑆 ↔ 𝐹 : 𝑆 –1-1-onto→ 𝑆 ) )
11		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑥 ) )
12		fveq1	⊢ ( 𝑓 = 𝐹 → ( 𝑓 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
13	11 12	sseq12d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 ‘ 𝑥 ) ⊆ ( 𝑓 ‘ 𝑦 ) ↔ ( 𝐹 ‘ 𝑥 ) ⊆ ( 𝐹 ‘ 𝑦 ) ) )
14	13	bibi2d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑥 ⊆ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ⊆ ( 𝑓 ‘ 𝑦 ) ) ↔ ( 𝑥 ⊆ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ⊆ ( 𝐹 ‘ 𝑦 ) ) ) )
15	14	2ralbidv	⊢ ( 𝑓 = 𝐹 → ( ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ⊆ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ⊆ ( 𝑓 ‘ 𝑦 ) ) ↔ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ⊆ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ⊆ ( 𝐹 ‘ 𝑦 ) ) ) )
16	10 15	anbi12d	⊢ ( 𝑓 = 𝐹 → ( ( 𝑓 : 𝑆 –1-1-onto→ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ⊆ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ⊆ ( 𝑓 ‘ 𝑦 ) ) ) ↔ ( 𝐹 : 𝑆 –1-1-onto→ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ⊆ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ⊆ ( 𝐹 ‘ 𝑦 ) ) ) ) )
17	9 16	elab3	⊢ ( 𝐹 ∈ { 𝑓 ∣ ( 𝑓 : 𝑆 –1-1-onto→ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ⊆ 𝑦 ↔ ( 𝑓 ‘ 𝑥 ) ⊆ ( 𝑓 ‘ 𝑦 ) ) ) } ↔ ( 𝐹 : 𝑆 –1-1-onto→ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ⊆ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ⊆ ( 𝐹 ‘ 𝑦 ) ) ) )
18	4 17	bitrdi	⊢ ( 𝐾 ∈ 𝐵 → ( 𝐹 ∈ 𝑀 ↔ ( 𝐹 : 𝑆 –1-1-onto→ 𝑆 ∧ ∀ 𝑥 ∈ 𝑆 ∀ 𝑦 ∈ 𝑆 ( 𝑥 ⊆ 𝑦 ↔ ( 𝐹 ‘ 𝑥 ) ⊆ ( 𝐹 ‘ 𝑦 ) ) ) ) )