Theorem subsfn 28034

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 28032	. 2 ⊢ -s = (𝑥 ∈ No , 𝑦 ∈ No ↦ (𝑥 +s ( -us ‘𝑦)))
2		ovex 7389	. 2 ⊢ (𝑥 +s ( -us ‘𝑦)) ∈ V
3	1, 2	fnmpoi 8012	1 ⊢ -s Fn ( No × No )

Syntax hints:   × cxp 5616   Fn wfn 6480  ‘cfv 6485  (class class class)co 7356   No csur 27621   +_s cadds 27969   -u_s cnegs 28029   -_s csubs 28030

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-subs 28032

This theorem is referenced by:  zsex  28390  elzs  28394