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 = ( 𝑅s { 𝐼 } )
pwssnf1o.b 𝐵 = ( Base ‘ 𝑅 )
pwssnf1o.f 𝐹 = ( 𝑥𝐵 ↦ ( { 𝐼 } × { 𝑥 } ) )
pwssnf1o.c 𝐶 = ( Base ‘ 𝑌 )
Assertion pwssnf1o ( ( 𝑅𝑉𝐼𝑊 ) → 𝐹 : 𝐵1-1-onto𝐶 )

Proof

Step Hyp Ref Expression
1 pwssnf1o.y 𝑌 = ( 𝑅s { 𝐼 } )
2 pwssnf1o.b 𝐵 = ( Base ‘ 𝑅 )
3 pwssnf1o.f 𝐹 = ( 𝑥𝐵 ↦ ( { 𝐼 } × { 𝑥 } ) )
4 pwssnf1o.c 𝐶 = ( Base ‘ 𝑌 )
5 2 fvexi 𝐵 ∈ V
6 simpr ( ( 𝑅𝑉𝐼𝑊 ) → 𝐼𝑊 )
7 3 mapsnf1o ( ( 𝐵 ∈ V ∧ 𝐼𝑊 ) → 𝐹 : 𝐵1-1-onto→ ( 𝐵m { 𝐼 } ) )
8 5 6 7 sylancr ( ( 𝑅𝑉𝐼𝑊 ) → 𝐹 : 𝐵1-1-onto→ ( 𝐵m { 𝐼 } ) )
9 snex { 𝐼 } ∈ V
10 1 2 pwsbas ( ( 𝑅𝑉 ∧ { 𝐼 } ∈ V ) → ( 𝐵m { 𝐼 } ) = ( Base ‘ 𝑌 ) )
11 9 10 mpan2 ( 𝑅𝑉 → ( 𝐵m { 𝐼 } ) = ( Base ‘ 𝑌 ) )
12 11 adantr ( ( 𝑅𝑉𝐼𝑊 ) → ( 𝐵m { 𝐼 } ) = ( Base ‘ 𝑌 ) )
13 4 12 eqtr4id ( ( 𝑅𝑉𝐼𝑊 ) → 𝐶 = ( 𝐵m { 𝐼 } ) )
14 13 f1oeq3d ( ( 𝑅𝑉𝐼𝑊 ) → ( 𝐹 : 𝐵1-1-onto𝐶𝐹 : 𝐵1-1-onto→ ( 𝐵m { 𝐼 } ) ) )
15 8 14 mpbird ( ( 𝑅𝑉𝐼𝑊 ) → 𝐹 : 𝐵1-1-onto𝐶 )