Theorem subsfo 28021

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 28020	. 2 ⊢ -s :( No × No )⟶ No
2		0sno 27790	. . . . 5 ⊢ 0s ∈ No
3		opelxpi 5691	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 0s ∈ No ) → ⟨𝑥, 0s ⟩ ∈ ( No × No ))
4	2, 3	mpan2 691	. . . 4 ⊢ (𝑥 ∈ No → ⟨𝑥, 0s ⟩ ∈ ( No × No ))
5		subsval 28016	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ 0s ∈ No ) → (𝑥 -s 0s ) = (𝑥 +s ( -us ‘ 0s )))
6	2, 5	mpan2 691	. . . . 5 ⊢ (𝑥 ∈ No → (𝑥 -s 0s ) = (𝑥 +s ( -us ‘ 0s )))
7		negs0s 27984	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
8	7	oveq2i 7416	. . . . . 6 ⊢ (𝑥 +s ( -us ‘ 0s )) = (𝑥 +s 0s )
9		addsrid 27923	. . . . . 6 ⊢ (𝑥 ∈ No → (𝑥 +s 0s ) = 𝑥)
10	8, 9	eqtrid 2782	. . . . 5 ⊢ (𝑥 ∈ No → (𝑥 +s ( -us ‘ 0s )) = 𝑥)
11	6, 10	eqtr2d 2771	. . . 4 ⊢ (𝑥 ∈ No → 𝑥 = (𝑥 -s 0s ))
12		fveq2 6876	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, 0s ⟩ → ( -s ‘𝑦) = ( -s ‘⟨𝑥, 0s ⟩))
13		df-ov 7408	. . . . . 6 ⊢ (𝑥 -s 0s ) = ( -s ‘⟨𝑥, 0s ⟩)
14	12, 13	eqtr4di 2788	. . . . 5 ⊢ (𝑦 = ⟨𝑥, 0s ⟩ → ( -s ‘𝑦) = (𝑥 -s 0s ))
15	14	rspceeqv 3624	. . . 4 ⊢ ((⟨𝑥, 0s ⟩ ∈ ( No × No ) ∧ 𝑥 = (𝑥 -s 0s )) → ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦))
16	4, 11, 15	syl2anc 584	. . 3 ⊢ (𝑥 ∈ No → ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦))
17	16	rgen 3053	. 2 ⊢ ∀𝑥 ∈ No ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦)
18		dffo3 7092	. 2 ⊢ ( -s :( No × No )–onto→ No ↔ ( -s :( No × No )⟶ No ∧ ∀𝑥 ∈ No ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦)))
19	1, 17, 18	mpbir2an 711	1 ⊢ -s :( No × No )–onto→ No

Syntax hints:   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060  ⟨cop 4607   × cxp 5652  ⟶wf 6527  –onto→wfo 6529  ‘cfv 6531  (class class class)co 7405   No csur 27603   0_s c0s 27786   +_s cadds 27918   -u_s cnegs 27977   -_s csubs 27978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3359  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-tp 4606  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-se 5607  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-pred 6290  df-ord 6355  df-on 6356  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7362  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-1o 8480  df-2o 8481  df-nadd 8678  df-no 27606  df-slt 27607  df-bday 27608  df-sslt 27745  df-scut 27747  df-0s 27788  df-made 27807  df-old 27808  df-left 27810  df-right 27811  df-norec 27897  df-norec2 27908  df-adds 27919  df-negs 27979  df-subs 27980

This theorem is referenced by:  zssno  28321