Theorem addsfo 28370

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 28369	. 2 ⊢ +s :(No × No)⟶No
2		0no 28196	. . . . 5 ⊢ 0s ∈ No
3		opelxpi 5687	. . . . 5 ⊢ ((𝑧 ∈ No ∧ 0s ∈ No) → ⟨𝑧, 0s ⟩ ∈ (No × No))
4	2, 3	mpan2 704	. . . 4 ⊢ (𝑧 ∈ No → ⟨𝑧, 0s ⟩ ∈ (No × No))
5		addsrid 28351	. . . . 5 ⊢ (𝑧 ∈ No → (𝑧 +s 0s ) = 𝑧)
6	5	eqcomd 2766	. . . 4 ⊢ (𝑧 ∈ No → 𝑧 = (𝑧 +s 0s ))
7		fveq2 6885	. . . . . 6 ⊢ (𝑥 = ⟨𝑧, 0s ⟩ → ( +s ‘𝑥) = ( +s ‘⟨𝑧, 0s ⟩))
8		df-ov 7423	. . . . . 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 7102	. 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 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086  ⟨cop 4589   × cxp 5648  ⟶wf 6534  –onto→wfo 6536  ‘cfv 6538  (class class class)co 7420  Nocsur 27998   0_s c0s 28192   +_s cadds 28346

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 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 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 8300  df-wrecs 8331  df-recs 8380  df-1o 8477  df-2o 8478  df-nadd 8676  df-no 28001  df-lts 28002  df-bday 28003  df-slts 28145  df-cuts 28147  df-0s 28194  df-made 28214  df-old 28215  df-left 28217  df-right 28218  df-norec2 28336  df-adds 28347

This theorem is used by: (None)