Theorem carsgmon 31572

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 5227	. 2 ⊢ (𝜑 → 𝐴 ∈ V)
4		id 22	. 2 ⊢ (𝜑 → 𝜑)
5		sseq1 3991	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝑦))
6	5	3anbi2d 1437	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝜑 ∧ 𝑥 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) ↔ (𝜑 ∧ 𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂)))
7		fveq2 6669	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑀‘𝑥) = (𝑀‘𝐴))
8	7	breq1d 5075	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑀‘𝑥) ≤ (𝑀‘𝑦) ↔ (𝑀‘𝐴) ≤ (𝑀‘𝑦)))
9	6, 8	imbi12d 347	. . . 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 3992	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝐴 ⊆ 𝑦 ↔ 𝐴 ⊆ 𝐵))
11		eleq1 2900	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ 𝒫 𝑂 ↔ 𝐵 ∈ 𝒫 𝑂))
12	10, 11	3anbi23d 1435	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝜑 ∧ 𝐴 ⊆ 𝑦 ∧ 𝑦 ∈ 𝒫 𝑂) ↔ (𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂)))
13		fveq2 6669	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑀‘𝑦) = (𝑀‘𝐵))
14	13	breq2d 5077	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑀‘𝐴) ≤ (𝑀‘𝑦) ↔ (𝑀‘𝐴) ≤ (𝑀‘𝐵)))
15	12, 14	imbi12d 347	. . . 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 3571	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) → ((𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂) → (𝑀‘𝐴) ≤ (𝑀‘𝐵)))
18	17	imp 409	. 2 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ 𝒫 𝑂) ∧ (𝜑 ∧ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝒫 𝑂)) → (𝑀‘𝐴) ≤ (𝑀‘𝐵))
19	3, 1, 4, 2, 1, 18	syl23anc 1373	1 ⊢ (𝜑 → (𝑀‘𝐴) ≤ (𝑀‘𝐵))

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  Vcvv 3494   ⊆ wss 3935  𝒫 cpw 4538   class class class wbr 5065  ⟶wf 6350  ‘cfv 6354  (class class class)co 7155  0cc0 10536  +∞cpnf 10671   ≤ cle 10675  [,]cicc 12740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-iota 6313  df-fv 6362

This theorem is referenced by:  carsggect  31576  carsgclctunlem2  31577