Theorem subsfo 28071

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 28070	. 2 ⊢ -s :( No × No )⟶ No
2		0no 27815	. . . . 5 ⊢ 0s ∈ No
3		opelxpi 5661	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 0s ∈ No ) → ⟨𝑥, 0s ⟩ ∈ ( No × No ))
4	2, 3	mpan2 692	. . . 4 ⊢ (𝑥 ∈ No → ⟨𝑥, 0s ⟩ ∈ ( No × No ))
5		subsval 28066	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ 0s ∈ No ) → (𝑥 -s 0s ) = (𝑥 +s ( -us ‘ 0s )))
6	2, 5	mpan2 692	. . . . 5 ⊢ (𝑥 ∈ No → (𝑥 -s 0s ) = (𝑥 +s ( -us ‘ 0s )))
7		neg0s 28032	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
8	7	oveq2i 7371	. . . . . 6 ⊢ (𝑥 +s ( -us ‘ 0s )) = (𝑥 +s 0s )
9		addsrid 27970	. . . . . 6 ⊢ (𝑥 ∈ No → (𝑥 +s 0s ) = 𝑥)
10	8, 9	eqtrid 2784	. . . . 5 ⊢ (𝑥 ∈ No → (𝑥 +s ( -us ‘ 0s )) = 𝑥)
11	6, 10	eqtr2d 2773	. . . 4 ⊢ (𝑥 ∈ No → 𝑥 = (𝑥 -s 0s ))
12		fveq2 6834	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, 0s ⟩ → ( -s ‘𝑦) = ( -s ‘⟨𝑥, 0s ⟩))
13		df-ov 7363	. . . . . 6 ⊢ (𝑥 -s 0s ) = ( -s ‘⟨𝑥, 0s ⟩)
14	12, 13	eqtr4di 2790	. . . . 5 ⊢ (𝑦 = ⟨𝑥, 0s ⟩ → ( -s ‘𝑦) = (𝑥 -s 0s ))
15	14	rspceeqv 3588	. . . 4 ⊢ ((⟨𝑥, 0s ⟩ ∈ ( No × No ) ∧ 𝑥 = (𝑥 -s 0s )) → ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦))
16	4, 11, 15	syl2anc 585	. . 3 ⊢ (𝑥 ∈ No → ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦))
17	16	rgen 3054	. 2 ⊢ ∀𝑥 ∈ No ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦)
18		dffo3 7048	. 2 ⊢ ( -s :( No × No )–onto→ No ↔ ( -s :( No × No )⟶ No ∧ ∀𝑥 ∈ No ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦)))
19	1, 17, 18	mpbir2an 712	1 ⊢ -s :( No × No )–onto→ No

Syntax hints:   = wceq 1542   ∈ wcel 2114  ∀wral 3052  ∃wrex 3062  ⟨cop 4574   × cxp 5622  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  (class class class)co 7360   No csur 27617   0_s c0s 27811   +_s cadds 27965   -u_s cnegs 28025   -_s csubs 28026

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-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  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-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5519  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-se 5578  df-we 5579  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  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 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-frecs 8224  df-wrecs 8255  df-recs 8304  df-1o 8398  df-2o 8399  df-nadd 8595  df-no 27620  df-lts 27621  df-bday 27622  df-slts 27764  df-cuts 27766  df-0s 27813  df-made 27833  df-old 27834  df-left 27836  df-right 27837  df-norec 27944  df-norec2 27955  df-adds 27966  df-negs 28027  df-subs 28028

This theorem is referenced by:  zssno  28387