Theorem cutminmax 28140

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 28066	. . . 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 5112	. . . . 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 28139	. 2 ⊢ (𝜑 → (( L ‘𝑋) |s ( R ‘𝑋)) = (( L ‘𝑋) |s {𝑅}))
9		cutminmax.1	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ ( L ‘𝑋))
10		elfvdm 6915	. . . . 5 ⊢ (𝐿 ∈ ( L ‘𝑋) → 𝑋 ∈ dom L )
11	9, 10	syl 18	. . . 4 ⊢ (𝜑 → 𝑋 ∈ dom L )
12		leftf 28059	. . . . 5 ⊢ L : No ⟶𝒫 No
13	12	fdmi 6717	. . . 4 ⊢ dom L = No
14	11, 13	eleqtrdi 2872	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
15		lrcut 28108	. . 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 4751	. . . 4 ⊢ (𝜑 → {𝑅} ⊆ ( R ‘𝑋))
18		ssslts2 27978	. . . 4 ⊢ ((( L ‘𝑋) <<s ( R ‘𝑋) ∧ {𝑅} ⊆ ( R ‘𝑋)) → ( L ‘𝑋) <<s {𝑅})
19	1, 17, 18	sylancr 598	. . 3 ⊢ (𝜑 → ( L ‘𝑋) <<s {𝑅})
20		cutminmax.2	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ( L ‘𝑋)𝑥 ≤s 𝐿)
21		breq1 5111	. . . . 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 28138	. 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 1569   ∈ wcel 2142  ∀wral 3078   ⊆ wss 3904  𝒫 cpw 4561  {csn 4588   class class class wbr 5108  dom cdm 5660  ‘cfv 6536  (class class class)co 7412   No csur 27815   ≤s cles 27919   <<s cslts 27961   |s ccuts 27963   L cleft 28029   R cright 28030

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  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 3368  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4595  df-uni 4872  df-int 4912  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-1o 8451  df-2o 8452  df-no 27818  df-lts 27819  df-bday 27820  df-les 27920  df-slts 27962  df-cuts 27964  df-made 28031  df-old 28032  df-left 28034  df-right 28035

This theorem is used by:  bdayfinbndlem1  28671