Theorem cutsfo 28273

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

Assertion

Ref	Expression
cutsfo	⊢ |s : <<s –onto→ No

Proof of Theorem cutsfo

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

Step	Hyp	Ref	Expression
1		cutsf 28160	. 2 ⊢ |s : <<s ⟶ No
2		lltr 28230	. . . . 5 ⊢ ( L ‘𝑥) <<s ( R ‘𝑥)
3		df-br 5104	. . . . 5 ⊢ (( L ‘𝑥) <<s ( R ‘𝑥) ↔ ⟨( L ‘𝑥), ( R ‘𝑥)⟩ ∈ <<s )
4	2, 3	mpbi 233	. . . 4 ⊢ ⟨( L ‘𝑥), ( R ‘𝑥)⟩ ∈ <<s
5		lrcut 28272	. . . . 5 ⊢ (𝑥 ∈ No → (( L ‘𝑥) |s ( R ‘𝑥)) = 𝑥)
6	5	eqcomd 2767	. . . 4 ⊢ (𝑥 ∈ No → 𝑥 = (( L ‘𝑥) |s ( R ‘𝑥)))
7		fveq2 6877	. . . . . 6 ⊢ (𝑦 = ⟨( L ‘𝑥), ( R ‘𝑥)⟩ → ( |s ‘𝑦) = ( |s ‘⟨( L ‘𝑥), ( R ‘𝑥)⟩))
8		df-ov 7415	. . . . . 6 ⊢ (( L ‘𝑥) |s ( R ‘𝑥)) = ( |s ‘⟨( L ‘𝑥), ( R ‘𝑥)⟩)
9	7, 8	eqtr4di 2814	. . . . 5 ⊢ (𝑦 = ⟨( L ‘𝑥), ( R ‘𝑥)⟩ → ( |s ‘𝑦) = (( L ‘𝑥) |s ( R ‘𝑥)))
10	9	rspceeqv 3599	. . . 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 3079	. 2 ⊢ ∀𝑥 ∈ No ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦)
13		dffo3 7094	. 2 ⊢ ( |s : <<s –onto→ No ↔ ( |s : <<s ⟶ No ∧ ∀𝑥 ∈ No ∃𝑦 ∈ <<s 𝑥 = ( |s ‘𝑦)))
14	1, 12, 13	mpbir2an 724	1 ⊢ |s : <<s –onto→ No

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  ⟨cop 4590   class class class wbr 5103  ⟶wf 6527  –onto→wfo 6529  ‘cfv 6531  (class class class)co 7412   No csur 27979   <<s cslts 28125   |s ccuts 28127   L cleft 28193   R cright 28194

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 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6297  df-ord 6358  df-on 6359  df-suc 6361  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-2nd 7991  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-1o 8460  df-2o 8461  df-no 27982  df-lts 27983  df-bday 27984  df-slts 28126  df-cuts 28128  df-made 28195  df-old 28196  df-left 28198  df-right 28199

This theorem is used by: (None)