Theorem cutminmax 27936

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 27862	. . . 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 5103	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑅 ≤s 𝑦 ↔ 𝑅 ≤s 𝑏))
6	5	cbvralvw 3215	. . . 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 27935	. 2 ⊢ (𝜑 → (( L ‘𝑋) |s ( R ‘𝑋)) = (( L ‘𝑋) |s {𝑅}))
9		cutminmax.1	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ ( L ‘𝑋))
10		elfvdm 6869	. . . . 5 ⊢ (𝐿 ∈ ( L ‘𝑋) → 𝑋 ∈ dom L )
11	9, 10	syl 17	. . . 4 ⊢ (𝜑 → 𝑋 ∈ dom L )
12		leftf 27855	. . . . 5 ⊢ L : No ⟶𝒫 No
13	12	fdmi 6674	. . . 4 ⊢ dom L = No
14	11, 13	eleqtrdi 2847	. . 3 ⊢ (𝜑 → 𝑋 ∈ No )
15		lrcut 27904	. . 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 4766	. . . 4 ⊢ (𝜑 → {𝑅} ⊆ ( R ‘𝑋))
18		ssslts2 27774	. . . 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 5102	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝑥 ≤s 𝐿 ↔ 𝑎 ≤s 𝐿))
22	21	cbvralvw 3215	. . . 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 27934	. 2 ⊢ (𝜑 → (( L ‘𝑋) |s {𝑅}) = ({𝐿} |s {𝑅}))
25	8, 16, 24	3eqtr3d 2780	1 ⊢ (𝜑 → 𝑋 = ({𝐿} |s {𝑅}))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3052   ⊆ wss 3902  𝒫 cpw 4555  {csn 4581   class class class wbr 5099  dom cdm 5625  ‘cfv 6493  (class class class)co 7360   No csur 27611   ≤s cles 27716   <<s cslts 27757   |s ccuts 27759   L cleft 27825   R cright 27826

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 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rmo 3351  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-tp 4586  df-op 4588  df-uni 4865  df-int 4904  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-ord 6321  df-on 6322  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7317  df-ov 7363  df-oprab 7364  df-mpo 7365  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-1o 8399  df-2o 8400  df-no 27614  df-lts 27615  df-bday 27616  df-les 27717  df-slts 27758  df-cuts 27760  df-made 27827  df-old 27828  df-left 27830  df-right 27831

This theorem is referenced by:  bdayfinbndlem1  28467