Theorem subsfo 28003

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 28002	. 2 ⊢ -s :( No × No )⟶ No
2		0sno 27768	. . . . 5 ⊢ 0s ∈ No
3		opelxpi 5653	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 0s ∈ No ) → ⟨𝑥, 0s ⟩ ∈ ( No × No ))
4	2, 3	mpan2 691	. . . 4 ⊢ (𝑥 ∈ No → ⟨𝑥, 0s ⟩ ∈ ( No × No ))
5		subsval 27998	. . . . . 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 27966	. . . . . . 7 ⊢ ( -us ‘ 0s ) = 0s
8	7	oveq2i 7357	. . . . . 6 ⊢ (𝑥 +s ( -us ‘ 0s )) = (𝑥 +s 0s )
9		addsrid 27905	. . . . . 6 ⊢ (𝑥 ∈ No → (𝑥 +s 0s ) = 𝑥)
10	8, 9	eqtrid 2778	. . . . 5 ⊢ (𝑥 ∈ No → (𝑥 +s ( -us ‘ 0s )) = 𝑥)
11	6, 10	eqtr2d 2767	. . . 4 ⊢ (𝑥 ∈ No → 𝑥 = (𝑥 -s 0s ))
12		fveq2 6822	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, 0s ⟩ → ( -s ‘𝑦) = ( -s ‘⟨𝑥, 0s ⟩))
13		df-ov 7349	. . . . . 6 ⊢ (𝑥 -s 0s ) = ( -s ‘⟨𝑥, 0s ⟩)
14	12, 13	eqtr4di 2784	. . . . 5 ⊢ (𝑦 = ⟨𝑥, 0s ⟩ → ( -s ‘𝑦) = (𝑥 -s 0s ))
15	14	rspceeqv 3600	. . . 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 3049	. 2 ⊢ ∀𝑥 ∈ No ∃𝑦 ∈ ( No × No )𝑥 = ( -s ‘𝑦)
18		dffo3 7035	. 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 1541   ∈ wcel 2111  ∀wral 3047  ∃wrex 3056  ⟨cop 4582   × cxp 5614  ⟶wf 6477  –onto→wfo 6479  ‘cfv 6481  (class class class)co 7346   No csur 27576   0_s c0s 27764   +_s cadds 27900   -u_s cnegs 27959   -_s csubs 27960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5217  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-tp 4581  df-op 4583  df-uni 4860  df-int 4898  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-tr 5199  df-id 5511  df-eprel 5516  df-po 5524  df-so 5525  df-fr 5569  df-se 5570  df-we 5571  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-pred 6248  df-ord 6309  df-on 6310  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-1o 8385  df-2o 8386  df-nadd 8581  df-no 27579  df-slt 27580  df-bday 27581  df-sslt 27719  df-scut 27721  df-0s 27766  df-made 27786  df-old 27787  df-left 27789  df-right 27790  df-norec 27879  df-norec2 27890  df-adds 27901  df-negs 27961  df-subs 27962

This theorem is referenced by:  zssno  28303