Theorem cutminmax 28199

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 28125	. . . 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 5111	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑅 ≤s 𝑦 ↔ 𝑅 ≤s 𝑏))
6	5	cbvralvw 3242	. . . 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 28198	. 2 ⊢ (𝜑 → (( L ‘𝑋) |s ( R ‘𝑋)) = (( L ‘𝑋) |s {𝑅}))
9		cutminmax.1	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ ( L ‘𝑋))
10		elfvdm 6916	. . . . 5 ⊢ (𝐿 ∈ ( L ‘𝑋) → 𝑋 ∈ dom L )
11	9, 10	syl 18	. . . 4 ⊢ (𝜑 → 𝑋 ∈ dom L )
12		leftf 28118	. . . . 5 ⊢ L : No ⟶𝒫 No
13	12	fdmi 6718	. . . 4 ⊢ dom L = No
14	11, 13	eleqtrdi 2872	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
15		lrcut 28167	. . 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 4750	. . . 4 ⊢ (𝜑 → {𝑅} ⊆ ( R ‘𝑋))
18		ssslts2 28037	. . . 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 5110	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝑥 ≤s 𝐿 ↔ 𝑎 ≤s 𝐿))
22	21	cbvralvw 3242	. . . 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 28197	. 2 ⊢ (𝜑 → (( L ‘𝑋) |s {𝑅}) = ({𝐿} |s {𝑅}))
25	8, 16, 24	3eqtr3d 2805	1 ⊢ (𝜑 → 𝑋 = ({𝐿} |s {𝑅}))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ⊆ wss 3902  𝒫 cpw 4560  {csn 4587   class class class wbr 5107  dom cdm 5659  ‘cfv 6537  (class class class)co 7416   No csur 27874   ≤s cles 27978   <<s cslts 28020   |s ccuts 28022   L cleft 28088   R cright 28089

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 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402  ax-un 7739

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 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-tp 4592  df-op 4594  df-uni 4871  df-int 4911  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-tr 5217  df-id 5554  df-eprel 5559  df-po 5567  df-so 5568  df-fr 5612  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-ord 6364  df-on 6365  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8363  df-1o 8458  df-2o 8459  df-no 27877  df-lts 27878  df-bday 27879  df-les 27979  df-slts 28021  df-cuts 28023  df-made 28090  df-old 28091  df-left 28093  df-right 28094

This theorem is used by:  bdayfinbndlem1  28730