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Metamath Proof Explorer

Theorem pwssnf1o

Description: Triviality of singleton powers: set equipollence. (Contributed by Stefan O'Rear, 24-Jan-2015)

Ref Expression
Hypotheses pwssnf1o.y ⊢ 𝑌 = ( 𝑅 ↑s { 𝐼 } )
pwssnf1o.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
pwssnf1o.f ⊢ 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( { 𝐼 } × { 𝑥 } ) )
pwssnf1o.c ⊢ 𝐶 = ( Base ‘ 𝑌 )
Assertion pwssnf1o ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐶 )

Proof

Step Hyp Ref Expression
1 pwssnf1o.y ⊢ 𝑌 = ( 𝑅 ↑s { 𝐼 } )
2 pwssnf1o.b ⊢ 𝐵 = ( Base ‘ 𝑅 )
3 pwssnf1o.f ⊢ 𝐹 = ( 𝑥 ∈ 𝐵 ↦ ( { 𝐼 } × { 𝑥 } ) )
4 pwssnf1o.c ⊢ 𝐶 = ( Base ‘ 𝑌 )
5 2 fvexi ⊢ 𝐵 ∈ V
6 simpr ⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → 𝐼 ∈ 𝑊 )
7 3 mapsnf1o ⊢ ( ( 𝐵 ∈ V ∧ 𝐼 ∈ 𝑊 ) → 𝐹 : 𝐵 –1-1-onto→ ( 𝐵 ↑m { 𝐼 } ) )
8 5 6 7 sylancr ⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → 𝐹 : 𝐵 –1-1-onto→ ( 𝐵 ↑m { 𝐼 } ) )
9 snex ⊢ { 𝐼 } ∈ V
10 1 2 pwsbas ⊢ ( ( 𝑅 ∈ 𝑉 ∧ { 𝐼 } ∈ V ) → ( 𝐵 ↑m { 𝐼 } ) = ( Base ‘ 𝑌 ) )
11 9 10 mpan2 ⊢ ( 𝑅 ∈ 𝑉 → ( 𝐵 ↑m { 𝐼 } ) = ( Base ‘ 𝑌 ) )
12 11 adantr ⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → ( 𝐵 ↑m { 𝐼 } ) = ( Base ‘ 𝑌 ) )
13 4 12 eqtr4id ⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → 𝐶 = ( 𝐵 ↑m { 𝐼 } ) )
14 13 f1oeq3d ⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → ( 𝐹 : 𝐵 –1-1-onto→ 𝐶 ↔ 𝐹 : 𝐵 –1-1-onto→ ( 𝐵 ↑m { 𝐼 } ) ) )
15 8 14 mpbird ⊢ ( ( 𝑅 ∈ 𝑉 ∧ 𝐼 ∈ 𝑊 ) → 𝐹 : 𝐵 –1-1-onto→ 𝐶 )