Theorem carsgmon 32260

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 5251	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
4		id 22	. 2 ⊢ (𝜑 → 𝜑)
5		sseq1 3950	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑦))
6	5	3anbi2d 1439	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) ↔ (𝜑 ∧ 𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂)))
7		fveq2 6768	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑀‘𝑥) = (𝑀‘𝐴))
8	7	breq1d 5088	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑀‘𝑥) ≤ (𝑀‘𝑦) ↔ (𝑀‘𝐴) ≤ (𝑀‘𝑦)))
9	6, 8	imbi12d 344	. . . 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 3951	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵))
11		eleq1 2827	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝒫 𝑂 ↔ 𝐵 ∈ 𝒫 𝑂))
12	10, 11	3anbi23d 1437	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝜑 ∧ 𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) ↔ (𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂)))
13		fveq2 6768	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑀‘𝑦) = (𝑀‘𝐵))
14	13	breq2d 5090	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑀‘𝐴) ≤ (𝑀‘𝑦) ↔ (𝑀‘𝐴) ≤ (𝑀‘𝐵)))
15	12, 14	imbi12d 344	. . . 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 3508	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) → ((𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂) → (𝑀‘𝐴) ≤ (𝑀‘𝐵)))
18	17	imp 406	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) ∧ (𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂)) → (𝑀‘𝐴) ≤ (𝑀‘𝐵))
19	3, 1, 4, 2, 1, 18	syl23anc 1375	1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1541   ∈ wcel 2109  Vcvv 3430   ⊆ wss 3891  𝒫 cpw 4538   class class class wbr 5078  ⟶wf 6426  ‘cfv 6430  (class class class)co 7268  0cc0 10855  +∞cpnf 10990   ≤ cle 10994  [,]cicc 13064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710  ax-sep 5226

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-iota 6388  df-fv 6438

This theorem is referenced by:  carsggect  32264  carsgclctunlem2  32265