Theorem addsfo 28399

Description: Surreal addition is onto. (Contributed by Scott Fenton, 21-Jan-2025.)

Assertion

Ref	Expression
addsfo	⊢ +s :(No × No)⟶ontoNo

Proof of Theorem addsfo

Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addsf 28398	. 2 ⊢ +s :(No × No)⟶No
2		0no 28225	. . . . 5 ⊢ 0s ∈ No
3		opelxpi 5692	. . . . 5 ⊢ ((𝑧 ∈ No ∧ 0s ∈ No) → ⟨𝑧, 0s ⟩ ∈ (No × No))
4	2, 3	mpan2 704	. . . 4 ⊢ (𝑧 ∈ No → ⟨𝑧, 0s ⟩ ∈ (No × No))
5		addsrid 28380	. . . . 5 ⊢ (𝑧 ∈ No → (𝑧 +s 0s ) = 𝑧)
6	5	eqcomd 2769	. . . 4 ⊢ (𝑧 ∈ No → 𝑧 = (𝑧 +s 0s ))
7		fveq2 6908	. . . . . 6 ⊢ (𝑥 = ⟨𝑧, 0s ⟩ → ( +s ‘𝑥) = ( +s ‘⟨𝑧, 0s ⟩))
8		df-ov 7446	. . . . . 6 ⊢ (𝑧 +s 0s ) = ( +s ‘⟨𝑧, 0s ⟩)
9	7, 8	eqtr4di 2816	. . . . 5 ⊢ (𝑥 = ⟨𝑧, 0s ⟩ → ( +s ‘𝑥) = (𝑧 +s 0s ))
10	9	rspceeqv 3602	. . . 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 3082	. 2 ⊢ ∀𝑧 ∈ No ∃𝑥 ∈ (No × No)𝑧 = ( +s ‘𝑥)
13		dffo3 7126	. 2 ⊢ ( +s :(No × No)⟶ontoNo ↔ ( +s :(No × No)⟶No ∧ ∀𝑧 ∈ No ∃𝑥 ∈ (No × No)𝑧 = ( +s ‘𝑥)))
14	1, 12, 13	mpbir2an 724	1 ⊢ +s :(No × No)⟶ontoNo

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ⟨cop 4838   × cxp 5653  ⟶wf 6554  ⟶ontowfo 6556  ‘cfv 6558  (class class class)co 7443  Nocsur 28027   0_s c0s 28221   +_s cadds 28375

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4839  df-uni 4871  df-int 4911  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6322  df-ord 6383  df-on 6384  df-suc 6386  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-f1 6563  df-fo 6564  df-f1o 6565  df-fv 6566  df-riota 7400  df-ov 7446  df-oprab 7447  df-mpo 7448  df-1st 8025  df-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-1o 8504  df-2o 8505  df-nadd 8703  df-no 28030  df-lts 28031  df-bday 28032  df-slts 28174  df-cuts 28176  df-0s 28223  df-made 28243  df-old 28244  df-left 28246  df-right 28247  df-norec2 28365  df-adds 28376

This theorem is used by: (None)