Theorem cutminmax 28349

Description: If the left set of 𝑋 has a maximum and the right set of 𝑋 has a minimum, then 𝑋 is equal to the cut of the maximum and the minimum. (Contributed by Scott Fenton, 23-Feb-2026.)

Hypotheses

Ref	Expression
cutminmax.1	⊢ (𝜑 → 𝐿 ∈ (L‘𝑋))
cutminmax.2	⊢ (𝜑 → ∀𝑥 ∈ (L‘𝑋)𝑥 ≤s 𝐿)
cutminmax.3	⊢ (𝜑 → 𝑅 ∈ (R‘𝑋))
cutminmax.4	⊢ (𝜑 → ∀𝑦 ∈ (R‘𝑋)𝑅 ≤s 𝑦)

Assertion

Ref	Expression
cutminmax	⊢ (𝜑 → 𝑋 = ({𝐿} |s {𝑅}))

Distinct variable groups:   𝑥,𝑋   𝑦,𝑋   𝑥,𝐿   𝑦,𝑅

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝑅(𝑥)   𝐿(𝑦)

Proof of Theorem cutminmax

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		lltr 28275	. . . 4 ⊢ (L‘𝑋) <<s (R‘𝑋)
2	1	a1i 11	. . 3 ⊢ (𝜑 → (L‘𝑋) <<s (R‘𝑋))
3		cutminmax.3	. . 3 ⊢ (𝜑 → 𝑅 ∈ (R‘𝑋))
4		cutminmax.4	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ (R‘𝑋)𝑅 ≤s 𝑦)
5		breq2 5110	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑅 ≤s 𝑦 ↔ 𝑅 ≤s 𝑏))
6	5	cbvralvw 3244	. . . 4 ⊢ (∀𝑦 ∈ (R‘𝑋)𝑅 ≤s 𝑦 ↔ ∀𝑏 ∈ (R‘𝑋)𝑅 ≤s 𝑏)
7	4, 6	sylib 221	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ (R‘𝑋)𝑅 ≤s 𝑏)
8	2, 3, 7	cutmin 28348	. 2 ⊢ (𝜑 → ((L‘𝑋) |s (R‘𝑋)) = ((L‘𝑋) |s {𝑅}))
9		cutminmax.1	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ (L‘𝑋))
10		elfvdm 6939	. . . . 5 ⊢ (𝐿 ∈ (L‘𝑋) → 𝑋 ∈ dom L)
11	9, 10	syl 18	. . . 4 ⊢ (𝜑 → 𝑋 ∈ dom L)
12		leftf 28268	. . . . 5 ⊢ L:No⟶𝒫 No
13	12	fdmi 6740	. . . 4 ⊢ dom L = No
14	11, 13	eleqtrdi 2874	. . 3 ⊢ (𝜑 → 𝑋 ∈ No)
15		lrcut 28317	. . 3 ⊢ (𝑋 ∈ No → ((L‘𝑋) |s (R‘𝑋)) = 𝑋)
16	14, 15	syl 18	. 2 ⊢ (𝜑 → ((L‘𝑋) |s (R‘𝑋)) = 𝑋)
17	3	snssd 4747	. . . 4 ⊢ (𝜑 → {𝑅} ⊆ (R‘𝑋))
18		ssslts2 28187	. . . 4 ⊢ (((L‘𝑋) <<s (R‘𝑋) ∧ {𝑅} ⊆ (R‘𝑋)) → (L‘𝑋) <<s {𝑅})
19	1, 17, 18	sylancr 599	. . 3 ⊢ (𝜑 → (L‘𝑋) <<s {𝑅})
20		cutminmax.2	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ (L‘𝑋)𝑥 ≤s 𝐿)
21		breq1 5109	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝑥 ≤s 𝐿 ↔ 𝑎 ≤s 𝐿))
22	21	cbvralvw 3244	. . . 4 ⊢ (∀𝑥 ∈ (L‘𝑋)𝑥 ≤s 𝐿 ↔ ∀𝑎 ∈ (L‘𝑋)𝑎 ≤s 𝐿)
23	20, 22	sylib 221	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ (L‘𝑋)𝑎 ≤s 𝐿)
24	19, 9, 23	cutmax 28347	. 2 ⊢ (𝜑 → ((L‘𝑋) |s {𝑅}) = ({𝐿} |s {𝑅}))
25	8, 16, 24	3eqtr3d 2806	1 ⊢ (𝜑 → 𝑋 = ({𝐿} |s {𝑅}))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ∀wral 3080   ⊆ wss 3900  𝒫 cpw 4560  {csn 4588   class class class wbr 5106  dom cdm 5655  ‘cfv 6555  (class class class)co 7440  Nocsur 28024   ≤s cles 28128   <<s cslts 28170   |s ccuts 28172  Lcleft 28238  Rcright 28239

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 7772

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 6319  df-ord 6380  df-on 6381  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-1o 8501  df-2o 8502  df-no 28027  df-lts 28028  df-bday 28029  df-les 28129  df-slts 28171  df-cuts 28173  df-made 28240  df-old 28241  df-left 28243  df-right 28244

This theorem is used by:  bdayfinbndlem1  28880