Theorem subsfn 28032

Description: Surreal subtraction is a function over pairs of surreals. (Contributed by Scott Fenton, 22-Jan-2025.)

Assertion

Ref	Expression
subsfn	⊢ -s Fn ( No × No )

Proof of Theorem subsfn

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-subs 28030	. 2 ⊢ -s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦)))
2		ovex 7401	. 2 ⊢ (𝑥 +s ( -us ‘𝑦)) ∈ V
3	1, 2	fnmpoi 8024	1 ⊢ -s Fn ( No × No )

Syntax hints:   × cxp 5630   Fn wfn 6495  ‘cfv 6500  (class class class)co 7368   No csur 27619   +_s cadds 27967   -u_s cnegs 28027   -_s csubs 28028

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-subs 28030

This theorem is referenced by:  zsex  28388  elzs  28392