Theorem carsgmon 30376

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 4805	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
4		id 22	. 2 ⊢ (𝜑 → 𝜑)
5		sseq1 3626	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑦))
6	5	3anbi2d 1404	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) ↔ (𝜑 ∧ 𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂)))
7		fveq2 6191	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑀‘𝑥) = (𝑀‘𝐴))
8	7	breq1d 4663	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑀‘𝑥) ≤ (𝑀‘𝑦) ↔ (𝑀‘𝐴) ≤ (𝑀‘𝑦)))
9	6, 8	imbi12d 334	. . . 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 3627	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵))
11		eleq1 2689	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝒫 𝑂 ↔ 𝐵 ∈ 𝒫 𝑂))
12	10, 11	3anbi23d 1402	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝜑 ∧ 𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) ↔ (𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂)))
13		fveq2 6191	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑀‘𝑦) = (𝑀‘𝐵))
14	13	breq2d 4665	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑀‘𝐴) ≤ (𝑀‘𝑦) ↔ (𝑀‘𝐴) ≤ (𝑀‘𝐵)))
15	12, 14	imbi12d 334	. . . 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 3270	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) → ((𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂) → (𝑀‘𝐴) ≤ (𝑀‘𝐵)))
18	17	imp 445	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) ∧ (𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂)) → (𝑀‘𝐴) ≤ (𝑀‘𝐵))
19	3, 1, 4, 2, 1, 18	syl23anc 1333	1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574  𝒫 cpw 4158   class class class wbr 4653  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  0cc0 9936  +∞cpnf 10071   ≤ cle 10075  [,]cicc 12178

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  carsggect  30380  carsgclctunlem2  30381