Theorem addsfo 28031

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.

Step	Hyp	Ref	Expression
1		addsf 28030	. 2 ⊢ +s :( No × No )⟶ No
2		0sno 27886	. . . . 5 ⊢ 0s ∈ No
3		opelxpi 5726	. . . . 5 ⊢ ((𝑧 ∈ No ∧ 0s ∈ No ) → ⟨𝑧, 0s ⟩ ∈ ( No × No ))
4	2, 3	mpan2 691	. . . 4 ⊢ (𝑧 ∈ No → ⟨𝑧, 0s ⟩ ∈ ( No × No ))
5		addsrid 28012	. . . . 5 ⊢ (𝑧 ∈ No → (𝑧 +s 0s ) = 𝑧)
6	5	eqcomd 2741	. . . 4 ⊢ (𝑧 ∈ No → 𝑧 = (𝑧 +s 0s ))
7		fveq2 6907	. . . . . 6 ⊢ (𝑥 = ⟨𝑧, 0s ⟩ → ( +s ‘𝑥) = ( +s ‘⟨𝑧, 0s ⟩))
8		df-ov 7434	. . . . . 6 ⊢ (𝑧 +s 0s ) = ( +s ‘⟨𝑧, 0s ⟩)
9	7, 8	eqtr4di 2793	. . . . 5 ⊢ (𝑥 = ⟨𝑧, 0s ⟩ → ( +s ‘𝑥) = (𝑧 +s 0s ))
10	9	rspceeqv 3645	. . . 4 ⊢ ((⟨𝑧, 0s ⟩ ∈ ( No × No ) ∧ 𝑧 = (𝑧 +s 0s )) → ∃𝑥 ∈ ( No × No )𝑧 = ( +s ‘𝑥))
11	4, 6, 10	syl2anc 584	. . 3 ⊢ (𝑧 ∈ No → ∃𝑥 ∈ ( No × No )𝑧 = ( +s ‘𝑥))
12	11	rgen 3061	. 2 ⊢ ∀𝑧 ∈ No ∃𝑥 ∈ ( No × No )𝑧 = ( +s ‘𝑥)
13		dffo3 7122	. 2 ⊢ ( +s :( No × No )–onto→ No ↔ ( +s :( No × No )⟶ No ∧ ∀𝑧 ∈ No ∃𝑥 ∈ ( No × No )𝑧 = ( +s ‘𝑥)))
14	1, 12, 13	mpbir2an 711	1 ⊢ +s :( No × No )–onto→ No

Syntax hints:   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  ⟨cop 4637   × cxp 5687  ⟶wf 6559  –onto→wfo 6561  ‘cfv 6563  (class class class)co 7431   No csur 27699   0_s c0s 27882   +_s cadds 28007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-1o 8505  df-2o 8506  df-nadd 8703  df-no 27702  df-slt 27703  df-bday 27704  df-sslt 27841  df-scut 27843  df-0s 27884  df-made 27901  df-old 27902  df-left 27904  df-right 27905  df-norec2 27997  df-adds 28008

This theorem is referenced by: (None)