Theorem cutminmax 27916

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		lltropt 27852	. . . 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 5101	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑅 ≤s 𝑦 ↔ 𝑅 ≤s 𝑏))
6	5	cbvralvw 3213	. . . 4 ⊢ (∀𝑦 ∈ ( R ‘𝑋)𝑅 ≤s 𝑦 ↔ ∀𝑏 ∈ ( R ‘𝑋)𝑅 ≤s 𝑏)
7	4, 6	sylib 218	. . 3 ⊢ (𝜑 → ∀𝑏 ∈ ( R ‘𝑋)𝑅 ≤s 𝑏)
8	2, 3, 7	cutmin 27915	. 2 ⊢ (𝜑 → (( L ‘𝑋) |s ( R ‘𝑋)) = (( L ‘𝑋) |s {𝑅}))
9		cutminmax.1	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ ( L ‘𝑋))
10		elfvdm 6867	. . . . 5 ⊢ (𝐿 ∈ ( L ‘𝑋) → 𝑋 ∈ dom L )
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝑋 ∈ dom L )
12		leftf 27845	. . . . 5 ⊢ L : No ⟶𝒫 No
13	12	fdmi 6672	. . . 4 ⊢ dom L = No
14	11, 13	eleqtrdi 2845	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
15		lrcut 27884	. . 3 ⊢ (𝑋 ∈ No → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
16	14, 15	syl 17	. 2 ⊢ (𝜑 → (( L ‘𝑋) |s ( R ‘𝑋)) = 𝑋)
17	3	snssd 4764	. . . 4 ⊢ (𝜑 → {𝑅} ⊆ ( R ‘𝑋))
18		sssslt2 27772	. . . 4 ⊢ ((( L ‘𝑋) <<s ( R ‘𝑋) ∧ {𝑅} ⊆ ( R ‘𝑋)) → ( L ‘𝑋) <<s {𝑅})
19	1, 17, 18	sylancr 588	. . 3 ⊢ (𝜑 → ( L ‘𝑋) <<s {𝑅})
20		cutminmax.2	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ ( L ‘𝑋)𝑥 ≤s 𝐿)
21		breq1 5100	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝑥 ≤s 𝐿 ↔ 𝑎 ≤s 𝐿))
22	21	cbvralvw 3213	. . . 4 ⊢ (∀𝑥 ∈ ( L ‘𝑋)𝑥 ≤s 𝐿 ↔ ∀𝑎 ∈ ( L ‘𝑋)𝑎 ≤s 𝐿)
23	20, 22	sylib 218	. . 3 ⊢ (𝜑 → ∀𝑎 ∈ ( L ‘𝑋)𝑎 ≤s 𝐿)
24	19, 9, 23	cutmax 27914	. 2 ⊢ (𝜑 → (( L ‘𝑋) |s {𝑅}) = ({𝐿} |s {𝑅}))
25	8, 16, 24	3eqtr3d 2778	1 ⊢ (𝜑 → 𝑋 = ({𝐿} |s {𝑅}))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3050   ⊆ wss 3900  𝒫 cpw 4553  {csn 4579   class class class wbr 5097  dom cdm 5623  ‘cfv 6491  (class class class)co 7358   No csur 27609   ≤s csle 27714   <<s csslt 27755   |s cscut 27757   L cleft 27821   R cright 27822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rmo 3349  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-tp 4584  df-op 4586  df-uni 4863  df-int 4902  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-pred 6258  df-ord 6319  df-on 6320  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-2nd 7934  df-frecs 8223  df-wrecs 8254  df-recs 8303  df-1o 8397  df-2o 8398  df-no 27612  df-slt 27613  df-bday 27614  df-sle 27715  df-sslt 27756  df-scut 27758  df-made 27823  df-old 27824  df-left 27826  df-right 27827

This theorem is referenced by:  bdayfinbndlem1  28444