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Metamath Proof Explorer

Theorem ispautN

Description: The predicate "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 ) ) ) ) )