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 )