Theorem cutsfo 28321

Description: The surreal cut function is onto. (Contributed by Scott Fenton, 23-Aug-2024.)

Assertion

Ref	Expression
cutsfo	⊢ |s : <<s ⟶ontoNo

Proof of Theorem cutsfo

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

Step	Hyp	Ref	Expression
1		cutsf 28208	. 2 ⊢ |s : <<s ⟶No
2		lltr 28278	. . . . 5 ⊢ (L‘𝑥) <<s (R‘𝑥)
3		df-br 5107	. . . . 5 ⊢ ((L‘𝑥) <<s (R‘𝑥) ↔ ⟨(L‘𝑥), (R‘𝑥)⟩ ∈ <<s )
4	2, 3	mpbi 233	. . . 4 ⊢ ⟨(L‘𝑥), (R‘𝑥)⟩ ∈ <<s
5		lrcut 28320	. . . . 5 ⊢ (𝑥 ∈ No → ((L‘𝑥) |s (R‘𝑥)) = 𝑥)
6	5	eqcomd 2769	. . . 4 ⊢ (𝑥 ∈ No → 𝑥 = ((L‘𝑥) |s (R‘𝑥)))
7		fveq2 6908	. . . . . 6 ⊢ (𝑦 = ⟨(L‘𝑥), (R‘𝑥)⟩ → ( |s ‘𝑦) = ( |s ‘⟨(L‘𝑥), (R‘𝑥)⟩))
8		df-ov 7446	. . . . . 6 ⊢ ((L‘𝑥) |s (R‘𝑥)) = ( |s ‘⟨(L‘𝑥), (R‘𝑥)⟩)
9	7, 8	eqtr4di 2816	. . . . 5 ⊢ (𝑦 = ⟨(L‘𝑥), (R‘𝑥)⟩ → ( |s ‘𝑦) = ((L‘𝑥) |s (R‘𝑥)))
10	9	rspceeqv 3602	. . . 4 ⊢ ((⟨(L‘𝑥), (R‘𝑥)⟩ ∈ <<s ∧ 𝑥 = ((L‘𝑥) |s (R‘𝑥))) → ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦))
11	4, 6, 10	sylancr 599	. . 3 ⊢ (𝑥 ∈ No → ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦))
12	11	rgen 3082	. 2 ⊢ ∀𝑥 ∈ No ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦)
13		dffo3 7126	. 2 ⊢ ( |s : <<s ⟶ontoNo ↔ ( |s : <<s ⟶No ∧ ∀𝑥 ∈ No ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦)))
14	1, 12, 13	mpbir2an 724	1 ⊢ |s : <<s ⟶ontoNo

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090  ⟨cop 4838   class class class wbr 5106  ⟶wf 6554  ⟶ontowfo 6556  ‘cfv 6558  (class class class)co 7443  Nocsur 28027   <<s cslts 28173   |s ccuts 28175  Lcleft 28241  Rcright 28242

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-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-2nd 8026  df-frecs 8327  df-wrecs 8358  df-recs 8407  df-1o 8504  df-2o 8505  df-no 28030  df-lts 28031  df-bday 28032  df-slts 28174  df-cuts 28176  df-made 28243  df-old 28244  df-left 28246  df-right 28247

This theorem is used by: (None)