Theorem carsgmon 33313

Description: Utility lemma: Apply monotony. (Contributed by Thierry Arnoux, 29-May-2020.)

Hypotheses

Ref	Expression
carsgval.1	⊢ (𝜑 → 𝑂 ∈ 𝑉)
carsgval.2	⊢ (𝜑 → 𝑀:𝒫 𝑂⟶(0[,]+∞))
carsgmon.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
carsgmon.2	⊢ (𝜑 → 𝐵 ∈ 𝒫 𝑂)
carsgmon.3	⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))

Assertion

Ref	Expression
carsgmon	⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑦,𝐵   𝑥,𝑀,𝑦   𝑥,𝑂,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥,𝑦)

Proof of Theorem carsgmon

Step	Hyp	Ref	Expression
1		carsgmon.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝒫 𝑂)
2		carsgmon.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3	1, 2	ssexd 5325	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
4		id 22	. 2 ⊢ (𝜑 → 𝜑)
5		sseq1 4008	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑦))
6	5	3anbi2d 1442	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) ↔ (𝜑 ∧ 𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂)))
7		fveq2 6892	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑀‘𝑥) = (𝑀‘𝐴))
8	7	breq1d 5159	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑀‘𝑥) ≤ (𝑀‘𝑦) ↔ (𝑀‘𝐴) ≤ (𝑀‘𝑦)))
9	6, 8	imbi12d 345	. . . 4 ⊢ (𝑥 = 𝐴 → (((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦)) ↔ ((𝜑 ∧ 𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝐴) ≤ (𝑀‘𝑦))))
10		sseq2 4009	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵))
11		eleq1 2822	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝒫 𝑂 ↔ 𝐵 ∈ 𝒫 𝑂))
12	10, 11	3anbi23d 1440	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝜑 ∧ 𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) ↔ (𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂)))
13		fveq2 6892	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑀‘𝑦) = (𝑀‘𝐵))
14	13	breq2d 5161	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑀‘𝐴) ≤ (𝑀‘𝑦) ↔ (𝑀‘𝐴) ≤ (𝑀‘𝐵)))
15	12, 14	imbi12d 345	. . . 4 ⊢ (𝑦 = 𝐵 → (((𝜑 ∧ 𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝐴) ≤ (𝑀‘𝑦)) ↔ ((𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂) → (𝑀‘𝐴) ≤ (𝑀‘𝐵))))
16		carsgmon.3	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) → (𝑀‘𝑥) ≤ (𝑀‘𝑦))
17	9, 15, 16	vtocl2g 3563	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) → ((𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂) → (𝑀‘𝐴) ≤ (𝑀‘𝐵)))
18	17	imp 408	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) ∧ (𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂)) → (𝑀‘𝐴) ≤ (𝑀‘𝐵))
19	3, 1, 4, 2, 1, 18	syl23anc 1378	1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3475   ⊆ wss 3949  𝒫 cpw 4603   class class class wbr 5149  ⟶wf 6540  ‘cfv 6544  (class class class)co 7409  0cc0 11110  +∞cpnf 11245   ≤ cle 11249  [,]cicc 13327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552

This theorem is referenced by:  carsggect  33317  carsgclctunlem2  33318