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Theorem addsfo 28016
Description: Surreal addition is onto. (Contributed by Scott Fenton, 21-Jan-2025.)
Assertion
Ref Expression
addsfo +s :( No × No )–onto No

Proof of Theorem addsfo
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 addsf 28015 . 2 +s :( No × No )⟶ No
2 0sno 27871 . . . . 5 0s No
3 opelxpi 5722 . . . . 5 ((𝑧 No ∧ 0s No ) → ⟨𝑧, 0s ⟩ ∈ ( No × No ))
42, 3mpan2 691 . . . 4 (𝑧 No → ⟨𝑧, 0s ⟩ ∈ ( No × No ))
5 addsrid 27997 . . . . 5 (𝑧 No → (𝑧 +s 0s ) = 𝑧)
65eqcomd 2743 . . . 4 (𝑧 No 𝑧 = (𝑧 +s 0s ))
7 fveq2 6906 . . . . . 6 (𝑥 = ⟨𝑧, 0s ⟩ → ( +s𝑥) = ( +s ‘⟨𝑧, 0s ⟩))
8 df-ov 7434 . . . . . 6 (𝑧 +s 0s ) = ( +s ‘⟨𝑧, 0s ⟩)
97, 8eqtr4di 2795 . . . . 5 (𝑥 = ⟨𝑧, 0s ⟩ → ( +s𝑥) = (𝑧 +s 0s ))
109rspceeqv 3645 . . . 4 ((⟨𝑧, 0s ⟩ ∈ ( No × No ) ∧ 𝑧 = (𝑧 +s 0s )) → ∃𝑥 ∈ ( No × No )𝑧 = ( +s𝑥))
114, 6, 10syl2anc 584 . . 3 (𝑧 No → ∃𝑥 ∈ ( No × No )𝑧 = ( +s𝑥))
1211rgen 3063 . 2 𝑧 No 𝑥 ∈ ( No × No )𝑧 = ( +s𝑥)
13 dffo3 7122 . 2 ( +s :( No × No )–onto No ↔ ( +s :( No × No )⟶ No ∧ ∀𝑧 No 𝑥 ∈ ( No × No )𝑧 = ( +s𝑥)))
141, 12, 13mpbir2an 711 1 +s :( No × No )–onto No
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cop 4632   × cxp 5683  wf 6557  ontowfo 6559  cfv 6561  (class class class)co 7431   No csur 27684   0s c0s 27867   +s cadds 27992
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-1o 8506  df-2o 8507  df-nadd 8704  df-no 27687  df-slt 27688  df-bday 27689  df-sslt 27826  df-scut 27828  df-0s 27869  df-made 27886  df-old 27887  df-left 27889  df-right 27890  df-norec2 27982  df-adds 27993
This theorem is referenced by: (None)
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