Theorem addsfo 28303

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 28302	. 2 ⊢ +s :( No × No )⟶ No
2		0no 28129	. . . . 5 ⊢ 0s ∈ No
3		opelxpi 5684	. . . . 5 ⊢ ((𝑧 ∈ No ∧ 0s ∈ No ) → ⟨𝑧, 0s ⟩ ∈ ( No × No ))
4	2, 3	mpan2 704	. . . 4 ⊢ (𝑧 ∈ No → ⟨𝑧, 0s ⟩ ∈ ( No × No ))
5		addsrid 28284	. . . . 5 ⊢ (𝑧 ∈ No → (𝑧 +s 0s ) = 𝑧)
6	5	eqcomd 2766	. . . 4 ⊢ (𝑧 ∈ No → 𝑧 = (𝑧 +s 0s ))
7		fveq2 6873	. . . . . 6 ⊢ (𝑥 = ⟨𝑧, 0s ⟩ → ( +s ‘𝑥) = ( +s ‘⟨𝑧, 0s ⟩))
8		df-ov 7411	. . . . . 6 ⊢ (𝑧 +s 0s ) = ( +s ‘⟨𝑧, 0s ⟩)
9	7, 8	eqtr4di 2813	. . . . 5 ⊢ (𝑥 = ⟨𝑧, 0s ⟩ → ( +s ‘𝑥) = (𝑧 +s 0s ))
10	9	rspceeqv 3598	. . . 4 ⊢ ((⟨𝑧, 0s ⟩ ∈ ( No × No ) ∧ 𝑧 = (𝑧 +s 0s )) → ∃𝑥 ∈ ( No × No )𝑧 = ( +s ‘𝑥))
11	4, 6, 10	syl2anc 596	. . 3 ⊢ (𝑧 ∈ No → ∃𝑥 ∈ ( No × No )𝑧 = ( +s ‘𝑥))
12	11	rgen 3078	. 2 ⊢ ∀𝑧 ∈ No ∃𝑥 ∈ ( No × No )𝑧 = ( +s ‘𝑥)
13		dffo3 7090	. 2 ⊢ ( +s :( No × No )–onto→ No ↔ ( +s :( No × No )⟶ No ∧ ∀𝑧 ∈ No ∃𝑥 ∈ ( No × No )𝑧 = ( +s ‘𝑥)))
14	1, 12, 13	mpbir2an 724	1 ⊢ +s :( No × No )–onto→ No

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086  ⟨cop 4589   × cxp 5645  ⟶wf 6523  –onto→wfo 6525  ‘cfv 6527  (class class class)co 7408   No csur 27931   0_s c0s 28125   +_s cadds 28279

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7734

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 2100  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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5542  df-eprel 5547  df-po 5555  df-so 5556  df-fr 5600  df-se 5601  df-we 5602  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-pred 6293  df-ord 6354  df-on 6355  df-suc 6357  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-f1 6532  df-fo 6533  df-f1o 6534  df-fv 6535  df-riota 7365  df-ov 7411  df-oprab 7412  df-mpo 7413  df-1st 7984  df-2nd 7985  df-frecs 8277  df-wrecs 8308  df-recs 8357  df-1o 8454  df-2o 8455  df-nadd 8653  df-no 27934  df-lts 27935  df-bday 27936  df-slts 28078  df-cuts 28080  df-0s 28127  df-made 28147  df-old 28148  df-left 28150  df-right 28151  df-norec2 28269  df-adds 28280

This theorem is used by: (None)