Theorem subsfo 28481

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

Assertion

Ref	Expression
subsfo	⊢ −s :(No × No)⟶ontoNo

Proof of Theorem subsfo

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

Step	Hyp	Ref	Expression
1		subsf 28480	. 2 ⊢ −s :(No × No)⟶No
2		0no 28225	. . . . 5 ⊢ 0s ∈ No
3		opelxpi 5692	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 0s ∈ No) → ⟨𝑥, 0s ⟩ ∈ (No × No))
4	2, 3	mpan2 704	. . . 4 ⊢ (𝑥 ∈ No → ⟨𝑥, 0s ⟩ ∈ (No × No))
5		subsval 28476	. . . . . 6 ⊢ ((𝑥 ∈ No ∧ 0s ∈ No) → (𝑥 −s 0s ) = (𝑥 +s ( -s‘ 0s )))
6	2, 5	mpan2 704	. . . . 5 ⊢ (𝑥 ∈ No → (𝑥 −s 0s ) = (𝑥 +s ( -s‘ 0s )))
7		neg0s 28442	. . . . . . 7 ⊢ ( -s‘ 0s ) = 0s
8	7	oveq2i 7454	. . . . . 6 ⊢ (𝑥 +s ( -s‘ 0s )) = (𝑥 +s 0s )
9		addsrid 28380	. . . . . 6 ⊢ (𝑥 ∈ No → (𝑥 +s 0s ) = 𝑥)
10	8, 9	eqtrid 2810	. . . . 5 ⊢ (𝑥 ∈ No → (𝑥 +s ( -s‘ 0s )) = 𝑥)
11	6, 10	eqtr2d 2799	. . . 4 ⊢ (𝑥 ∈ No → 𝑥 = (𝑥 −s 0s ))
12		fveq2 6908	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, 0s ⟩ → ( −s ‘𝑦) = ( −s ‘⟨𝑥, 0s ⟩))
13		df-ov 7446	. . . . . 6 ⊢ (𝑥 −s 0s ) = ( −s ‘⟨𝑥, 0s ⟩)
14	12, 13	eqtr4di 2816	. . . . 5 ⊢ (𝑦 = ⟨𝑥, 0s ⟩ → ( −s ‘𝑦) = (𝑥 −s 0s ))
15	14	rspceeqv 3602	. . . 4 ⊢ ((⟨𝑥, 0s ⟩ ∈ (No × No) ∧ 𝑥 = (𝑥 −s 0s )) → ∃𝑦 ∈ (No × No)𝑥 = ( −s ‘𝑦))
16	4, 11, 15	syl2anc 596	. . 3 ⊢ (𝑥 ∈ No → ∃𝑦 ∈ (No × No)𝑥 = ( −s ‘𝑦))
17	16	rgen 3082	. 2 ⊢ ∀𝑥 ∈ No ∃𝑦 ∈ (No × No)𝑥 = ( −s ‘𝑦)
18		dffo3 7126	. 2 ⊢ ( −s :(No × No)⟶ontoNo ↔ ( −s :(No × No)⟶No ∧ ∀𝑥 ∈ No ∃𝑦 ∈ (No × No)𝑥 = ( −s ‘𝑦)))
19	1, 17, 18	mpbir2an 724	1 ⊢ −s :(No × No)⟶ontoNo

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ⟨cop 4838   × cxp 5653  ⟶wf 6554  ⟶ontowfo 6556  ‘cfv 6558  (class class class)co 7443  Nocsur 28027   0_s c0s 28221   +_s cadds 28375   -_scnegs 28435   −_s csubs 28436

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-1o 8504  df-2o 8505  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-slts 28174  df-cuts 28176  df-0s 28223  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec 28354  df-norec2 28365  df-adds 28376  df-negs 28437  df-subs 28438

This theorem is used by:  zssno  28797