Theorem negsfo

Description: Function statement for surreal negation. (Contributed by Scott Fenton, 3-Feb-2025)

	Ref	Expression
Assertion	negsfo	⊢ -us : No –onto→ No

Proof

Step	Hyp	Ref	Expression
1		negsf	⊢ -us : No ⟶ No
2		negscl	⊢ ( 𝑥 ∈ No → ( -us ‘ 𝑥 ) ∈ No )
3		negnegs	⊢ ( 𝑥 ∈ No → ( -us ‘ ( -us ‘ 𝑥 ) ) = 𝑥 )
4	3	eqcomd	⊢ ( 𝑥 ∈ No → 𝑥 = ( -us ‘ ( -us ‘ 𝑥 ) ) )
5		fveq2	⊢ ( 𝑦 = ( -us ‘ 𝑥 ) → ( -us ‘ 𝑦 ) = ( -us ‘ ( -us ‘ 𝑥 ) ) )
6	5	eqeq2d	⊢ ( 𝑦 = ( -us ‘ 𝑥 ) → ( 𝑥 = ( -us ‘ 𝑦 ) ↔ 𝑥 = ( -us ‘ ( -us ‘ 𝑥 ) ) ) )
7	6	rspcev	⊢ ( ( ( -us ‘ 𝑥 ) ∈ No ∧ 𝑥 = ( -us ‘ ( -us ‘ 𝑥 ) ) ) → ∃ 𝑦 ∈ No 𝑥 = ( -us ‘ 𝑦 ) )
8	2 4 7	syl2anc	⊢ ( 𝑥 ∈ No → ∃ 𝑦 ∈ No 𝑥 = ( -us ‘ 𝑦 ) )
9	8	rgen	⊢ ∀ 𝑥 ∈ No ∃ 𝑦 ∈ No 𝑥 = ( -us ‘ 𝑦 )
10		dffo3	⊢ ( -us : No –onto→ No ↔ ( -us : No ⟶ No ∧ ∀ 𝑥 ∈ No ∃ 𝑦 ∈ No 𝑥 = ( -us ‘ 𝑦 ) ) )
11	1 9 10	mpbir2an	⊢ -us : No –onto→ No