Theorem subsfo 28082

Description: Surreal subtraction is an onto function. (Contributed by Scott Fenton, 17-May-2025.)

Assertion

Ref	Expression
subsfo	⊢ -s :( No × No )–onto→ No

Proof of Theorem subsfo

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

Step	Hyp	Ref	Expression
1		subsf 28081	. 2 ⊢ -s :( No × No )⟶ No
2		0no 27826	. . . . 5 ⊢ 0s ∈ No
3		opelxpi 5662	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 0s ∈ No ) → ⟨𝑥, 0s ⟩ ∈ ( No × No ))
4	2, 3	mpan2 697	. . . 4 ⊢ (𝑥 ∈ No → ⟨𝑥, 0s ⟩ ∈ ( No × No ))
5		subsval 28077	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ 0s ∈ No ) → (𝑥 -s 0s ) = (𝑥 +s ( -us ‘ 0s )))
6	2, 5	mpan2 697	. . . . 5 ⊢ (𝑥 ∈ No → (𝑥 -s 0s ) = (𝑥 +s ( -us ‘ 0s )))
7		neg0s 28043	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
8	7	oveq2i 7374	. . . . . 6 ⊢ (𝑥 +s ( -us ‘ 0s )) = (𝑥 +s 0s )
9		addsrid 27981	. . . . . 6 ⊢ (𝑥 ∈ No → (𝑥 +s 0s ) = 𝑥)
10	8, 9	eqtrid 2787	. . . . 5 ⊢ (𝑥 ∈ No → (𝑥 +s ( -us ‘ 0s )) = 𝑥)
11	6, 10	eqtr2d 2776	. . . 4 ⊢ (𝑥 ∈ No → 𝑥 = (𝑥 -s 0s ))
12		fveq2 6834	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, 0s ⟩ → ( -s ‘𝑦) = ( -s ‘⟨𝑥, 0s ⟩))
13		df-ov 7366	. . . . . 6 ⊢ (𝑥 -s 0s ) = ( -s ‘⟨𝑥, 0s ⟩)
14	12, 13	eqtr4di 2793	. . . . 5 ⊢ (𝑦 = ⟨𝑥, 0s ⟩ → ( -s ‘𝑦) = (𝑥 -s 0s ))
15	14	rspceeqv 3590	. . . 4 ⊢ ((⟨𝑥, 0s ⟩ ∈ ( No × No ) ∧ 𝑥 = (𝑥 -s 0s )) → ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦))
16	4, 11, 15	syl2anc 590	. . 3 ⊢ (𝑥 ∈ No → ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦))
17	16	rgen 3056	. 2 ⊢ ∀𝑥 ∈ No ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦)
18		dffo3 7050	. 2 ⊢ ( -s :( No × No )–onto→ No ↔ ( -s :( No × No )⟶ No ∧ ∀𝑥 ∈ No ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦)))
19	1, 17, 18	mpbir2an 717	1 ⊢ -s :( No × No )–onto→ No

Syntax hints:   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ∃wrex 3064  ⟨cop 4568   × cxp 5623  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7363   No csur 27628   0_s c0s 27822   +_s cadds 27976   -u_s cnegs 28036   -_s csubs 28037

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 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  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 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rmo 3345  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-tp 4567  df-op 4569  df-uni 4846  df-int 4885  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-tr 5187  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-se 5579  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6259  df-ord 6320  df-on 6321  df-suc 6323  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7320  df-ov 7366  df-oprab 7367  df-mpo 7368  df-1st 7938  df-2nd 7939  df-frecs 8228  df-wrecs 8259  df-recs 8308  df-1o 8402  df-2o 8403  df-nadd 8599  df-no 27631  df-lts 27632  df-bday 27633  df-slts 27775  df-cuts 27777  df-0s 27824  df-made 27844  df-old 27845  df-left 27847  df-right 27848  df-norec 27955  df-norec2 27966  df-adds 27977  df-negs 28038  df-subs 28039

This theorem is referenced by:  zssno  28398