Theorem isf32lem1 10306

Description: Lemma for isfin3-2 10320. Derive weak ordering property. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Hypotheses

Ref	Expression
isf32lem.a	⊢ (𝜑 → 𝐹:ω⟶𝒫 𝐺)
isf32lem.b	⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥))
isf32lem.c	⊢ (𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹)

Assertion

Ref	Expression
isf32lem1	⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝐴 ∧ 𝜑)) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵))

Distinct variable groups:   𝑥,𝐵   𝜑,𝑥   𝑥,𝐴   𝑥,𝐹

Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem isf32lem1

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6858	. . . . 5 ⊢ (𝑎 = 𝐵 → (𝐹‘𝑎) = (𝐹‘𝐵))
2	1	sseq1d 3978	. . . 4 ⊢ (𝑎 = 𝐵 → ((𝐹‘𝑎) ⊆ (𝐹‘𝐵) ↔ (𝐹‘𝐵) ⊆ (𝐹‘𝐵)))
3	2	imbi2d 340	. . 3 ⊢ (𝑎 = 𝐵 → ((𝜑 → (𝐹‘𝑎) ⊆ (𝐹‘𝐵)) ↔ (𝜑 → (𝐹‘𝐵) ⊆ (𝐹‘𝐵))))
4		fveq2 6858	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
5	4	sseq1d 3978	. . . 4 ⊢ (𝑎 = 𝑏 → ((𝐹‘𝑎) ⊆ (𝐹‘𝐵) ↔ (𝐹‘𝑏) ⊆ (𝐹‘𝐵)))
6	5	imbi2d 340	. . 3 ⊢ (𝑎 = 𝑏 → ((𝜑 → (𝐹‘𝑎) ⊆ (𝐹‘𝐵)) ↔ (𝜑 → (𝐹‘𝑏) ⊆ (𝐹‘𝐵))))
7		fveq2 6858	. . . . 5 ⊢ (𝑎 = suc 𝑏 → (𝐹‘𝑎) = (𝐹‘suc 𝑏))
8	7	sseq1d 3978	. . . 4 ⊢ (𝑎 = suc 𝑏 → ((𝐹‘𝑎) ⊆ (𝐹‘𝐵) ↔ (𝐹‘suc 𝑏) ⊆ (𝐹‘𝐵)))
9	8	imbi2d 340	. . 3 ⊢ (𝑎 = suc 𝑏 → ((𝜑 → (𝐹‘𝑎) ⊆ (𝐹‘𝐵)) ↔ (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹‘𝐵))))
10		fveq2 6858	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
11	10	sseq1d 3978	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑎) ⊆ (𝐹‘𝐵) ↔ (𝐹‘𝐴) ⊆ (𝐹‘𝐵)))
12	11	imbi2d 340	. . 3 ⊢ (𝑎 = 𝐴 → ((𝜑 → (𝐹‘𝑎) ⊆ (𝐹‘𝐵)) ↔ (𝜑 → (𝐹‘𝐴) ⊆ (𝐹‘𝐵))))
13		ssid 3969	. . . 4 ⊢ (𝐹‘𝐵) ⊆ (𝐹‘𝐵)
14	13	2a1i 12	. . 3 ⊢ (𝐵 ∈ ω → (𝜑 → (𝐹‘𝐵) ⊆ (𝐹‘𝐵)))
15		isf32lem.b	. . . . . . 7 ⊢ (𝜑 → ∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥))
16		suceq 6400	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → suc 𝑥 = suc 𝑏)
17	16	fveq2d 6862	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑏))
18		fveq2 6858	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (𝐹‘𝑥) = (𝐹‘𝑏))
19	17, 18	sseq12d 3980	. . . . . . . 8 ⊢ (𝑥 = 𝑏 → ((𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏)))
20	19	rspcv 3584	. . . . . . 7 ⊢ (𝑏 ∈ ω → (∀𝑥 ∈ ω (𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥) → (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏)))
21	15, 20	syl5 34	. . . . . 6 ⊢ (𝑏 ∈ ω → (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏)))
22	21	ad2antrr 726	. . . . 5 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑏) → (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏)))
23		sstr2 3953	. . . . 5 ⊢ ((𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏) → ((𝐹‘𝑏) ⊆ (𝐹‘𝐵) → (𝐹‘suc 𝑏) ⊆ (𝐹‘𝐵)))
24	22, 23	syl6 35	. . . 4 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑏) → (𝜑 → ((𝐹‘𝑏) ⊆ (𝐹‘𝐵) → (𝐹‘suc 𝑏) ⊆ (𝐹‘𝐵))))
25	24	a2d 29	. . 3 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑏) → ((𝜑 → (𝐹‘𝑏) ⊆ (𝐹‘𝐵)) → (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹‘𝐵))))
26	3, 6, 9, 12, 14, 25	findsg 7873	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (𝜑 → (𝐹‘𝐴) ⊆ (𝐹‘𝐵)))
27	26	impr 454	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝐴 ∧ 𝜑)) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3914  𝒫 cpw 4563  ∩ cint 4910  ran crn 5639  suc csuc 6334  ⟶wf 6507  ‘cfv 6511  ωcom 7842

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-tr 5215  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-ord 6335  df-on 6336  df-lim 6337  df-suc 6338  df-iota 6464  df-fv 6519  df-om 7843

This theorem is referenced by:  isf32lem2  10307  isf32lem3  10308