Theorem cutminmax 28095

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 28021	. . . 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 5115	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑅 ≤s 𝑦 ↔ 𝑅 ≤s 𝑏))
6	5	cbvralvw 3249	. . . 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 28094	. 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 28014	. . . . 5 ⊢ L : No ⟶𝒫 No
13	12	fdmi 6718	. . . 4 ⊢ dom L = No
14	11, 13	eleqtrdi 2879	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
15		lrcut 28063	. . 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 4755	. . . 4 ⊢ (𝜑 → {𝑅} ⊆ ( R ‘𝑋))
18		ssslts2 27933	. . . 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 5114	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝑥 ≤s 𝐿 ↔ 𝑎 ≤s 𝐿))
22	21	cbvralvw 3249	. . . 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 28093	. 2 ⊢ (𝜑 → (( L ‘𝑋) |s {𝑅}) = ({𝐿} |s {𝑅}))
25	8, 16, 24	3eqtr3d 2812	1 ⊢ (𝜑 → 𝑋 = ({𝐿} |s {𝑅}))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ⊆ wss 3911  𝒫 cpw 4565  {csn 4592   class class class wbr 5111  dom cdm 5662  ‘cfv 6537  (class class class)co 7411   No csur 27770   ≤s cles 27874   <<s cslts 27916   |s ccuts 27918   L cleft 27984   R cright 27985

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rmo 3375  df-reu 3376  df-rab 3423  df-v 3463  df-sbc 3752  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3931  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4915  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  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 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-2nd 7987  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-1o 8453  df-2o 8454  df-no 27773  df-lts 27774  df-bday 27775  df-les 27875  df-slts 27917  df-cuts 27919  df-made 27986  df-old 27987  df-left 27989  df-right 27990

This theorem is referenced by:  bdayfinbndlem1  28626