Theorem cutminmax 28322

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 28248	. . . 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 5106	. . . . 5 ⊢ (𝑦 = 𝑏 → (𝑅 ≤s 𝑦 ↔ 𝑅 ≤s 𝑏))
6	5	cbvralvw 3240	. . . 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 28321	. 2 ⊢ (𝜑 → ((L‘𝑋) |s (R‘𝑋)) = ((L‘𝑋) |s {𝑅}))
9		cutminmax.1	. . . . 5 ⊢ (𝜑 → 𝐿 ∈ (L‘𝑋))
10		elfvdm 6919	. . . . 5 ⊢ (𝐿 ∈ (L‘𝑋) → 𝑋 ∈ dom L)
11	9, 10	syl 18	. . . 4 ⊢ (𝜑 → 𝑋 ∈ dom L)
12		leftf 28241	. . . . 5 ⊢ L:No⟶𝒫 No
13	12	fdmi 6721	. . . 4 ⊢ dom L = No
14	11, 13	eleqtrdi 2870	. . 3 ⊢ (𝜑 → 𝑋 ∈ No)
15		lrcut 28290	. . 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 4746	. . . 4 ⊢ (𝜑 → {𝑅} ⊆ (R‘𝑋))
18		ssslts2 28160	. . . 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 5105	. . . . 5 ⊢ (𝑥 = 𝑎 → (𝑥 ≤s 𝐿 ↔ 𝑎 ≤s 𝐿))
22	21	cbvralvw 3240	. . . 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 28320	. 2 ⊢ (𝜑 → ((L‘𝑋) |s {𝑅}) = ({𝐿} |s {𝑅}))
25	8, 16, 24	3eqtr3d 2803	1 ⊢ (𝜑 → 𝑋 = ({𝐿} |s {𝑅}))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ⊆ wss 3895  𝒫 cpw 4556  {csn 4583   class class class wbr 5102  dom cdm 5650  ‘cfv 6538  (class class class)co 7420  Nocsur 27997   ≤s cles 28101   <<s cslts 28143   |s ccuts 28145  Lcleft 28211  Rcright 28212

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-tp 4588  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-ord 6365  df-on 6366  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-1o 8476  df-2o 8477  df-no 28000  df-lts 28001  df-bday 28002  df-les 28102  df-slts 28144  df-cuts 28146  df-made 28213  df-old 28214  df-left 28216  df-right 28217

This theorem is used by:  bdayfinbndlem1  28853