Theorem glble 18464

Description: The greatest lower bound is the least element. (Contributed by NM, 22-Oct-2011.) (Revised by NM, 7-Sep-2018.)

Hypotheses

Ref	Expression
glbprop.b	⊢ 𝐵 = (Base‘𝐾)
glbprop.l	⊢ ≤ = (le‘𝐾)
glbprop.u	⊢ 𝑈 = (glb‘𝐾)
glbprop.k	⊢ (𝜑 → 𝐾 ∈ 𝑉)
glbprop.s	⊢ (𝜑 → 𝑆 ∈ dom 𝑈)
glble.x	⊢ (𝜑 → 𝑋 ∈ 𝑆)

Assertion

Ref	Expression
glble	⊢ (𝜑 → (𝑈‘𝑆) ≤ 𝑋)

Proof of Theorem glble

Dummy variables 𝑧 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq2 5111	. 2 ⊢ (𝑦 = 𝑋 → ((𝑈‘𝑆) ≤ 𝑦 ↔ (𝑈‘𝑆) ≤ 𝑋))
2		glbprop.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
3		glbprop.l	. . . 4 ⊢ ≤ = (le‘𝐾)
4		glbprop.u	. . . 4 ⊢ 𝑈 = (glb‘𝐾)
5		glbprop.k	. . . 4 ⊢ (𝜑 → 𝐾 ∈ 𝑉)
6		glbprop.s	. . . 4 ⊢ (𝜑 → 𝑆 ∈ dom 𝑈)
7	2, 3, 4, 5, 6	glbprop 18463	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦 ∧ ∀𝑧 ∈ 𝐵 (∀𝑦 ∈ 𝑆 𝑧 ≤ 𝑦 → 𝑧 ≤ (𝑈‘𝑆))))
8	7	simpld 500	. 2 ⊢ (𝜑 → ∀𝑦 ∈ 𝑆 (𝑈‘𝑆) ≤ 𝑦)
9		glble.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝑆)
10	1, 8, 9	rspcdva 3580	1 ⊢ (𝜑 → (𝑈‘𝑆) ≤ 𝑋)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ∀wral 3078   class class class wbr 5107  dom cdm 5659  ‘cfv 6537  Basecbs 17307  lecple 17355  glbcglb 18404

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-rep 5236  ax-sep 5255  ax-nul 5267  ax-pow 5334  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3367  df-reu 3368  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-iun 4956  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  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 7374  df-glb 18439

This theorem is used by:  p0le  18521  clatglble  18611  glbsscl  49895