Theorem subsfo 28456

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 28455	. 2 ⊢ −s :(No × No)⟶No
2		0no 28200	. . . . 5 ⊢ 0s ∈ No
3		opelxpi 5687	. . . . 5 ⊢ ((𝑥 ∈ No ∧ 0s ∈ No) → ⟨𝑥, 0s ⟩ ∈ (No × No))
4	2, 3	mpan2 704	. . . 4 ⊢ (𝑥 ∈ No → ⟨𝑥, 0s ⟩ ∈ (No × No))
5		subsval 28451	. . . . . 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 28417	. . . . . . 7 ⊢ ( -s‘ 0s ) = 0s
8	7	oveq2i 7431	. . . . . 6 ⊢ (𝑥 +s ( -s‘ 0s )) = (𝑥 +s 0s )
9		addsrid 28355	. . . . . 6 ⊢ (𝑥 ∈ No → (𝑥 +s 0s ) = 𝑥)
10	8, 9	eqtrid 2807	. . . . 5 ⊢ (𝑥 ∈ No → (𝑥 +s ( -s‘ 0s )) = 𝑥)
11	6, 10	eqtr2d 2796	. . . 4 ⊢ (𝑥 ∈ No → 𝑥 = (𝑥 −s 0s ))
12		fveq2 6885	. . . . . 6 ⊢ (𝑦 = ⟨𝑥, 0s ⟩ → ( −s ‘𝑦) = ( −s ‘⟨𝑥, 0s ⟩))
13		df-ov 7423	. . . . . 6 ⊢ (𝑥 −s 0s ) = ( −s ‘⟨𝑥, 0s ⟩)
14	12, 13	eqtr4di 2813	. . . . 5 ⊢ (𝑦 = ⟨𝑥, 0s ⟩ → ( −s ‘𝑦) = (𝑥 −s 0s ))
15	14	rspceeqv 3598	. . . 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 3078	. 2 ⊢ ∀𝑥 ∈ No ∃𝑦 ∈ (No × No)𝑥 = ( −s ‘𝑦)
18		dffo3 7102	. 2 ⊢ ( −s :(No × No)–onto→No ↔ ( −s :(No × No)⟶No ∧ ∀𝑥 ∈ No ∃𝑦 ∈ (No × No)𝑥 = ( −s ‘𝑦)))
19	1, 17, 18	mpbir2an 724	1 ⊢ −s :(No × No)–onto→No

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  ⟨cop 4834   × cxp 5648  ⟶wf 6534  –onto→wfo 6536  ‘cfv 6538  (class class class)co 7420  Nocsur 28002   0_s c0s 28196   +_s cadds 28350   -_scnegs 28410   −_s csubs 28411

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-se 5604  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8302  df-wrecs 8333  df-recs 8382  df-1o 8479  df-2o 8480  df-nadd 8678  df-no 28005  df-lts 28006  df-bday 28007  df-slts 28149  df-cuts 28151  df-0s 28198  df-made 28218  df-old 28219  df-left 28221  df-right 28222  df-norec 28329  df-norec2 28340  df-adds 28351  df-negs 28412  df-subs 28413

This theorem is used by:  zssno  28772