Theorem isf32lem1 10422

Description: Lemma for isfin3-2 10436. 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 6920	. . . . 5 ⊢ (𝑎 = 𝐵 → (𝐹‘𝑎) = (𝐹‘𝐵))
2	1	sseq1d 4040	. . . 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 6920	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
5	4	sseq1d 4040	. . . 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 6920	. . . . 5 ⊢ (𝑎 = suc 𝑏 → (𝐹‘𝑎) = (𝐹‘suc 𝑏))
8	7	sseq1d 4040	. . . 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 6920	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
11	10	sseq1d 4040	. . . 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 4031	. . . 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 6461	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → suc 𝑥 = suc 𝑏)
17	16	fveq2d 6924	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑏))
18		fveq2 6920	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (𝐹‘𝑥) = (𝐹‘𝑏))
19	17, 18	sseq12d 4042	. . . . . . . 8 ⊢ (𝑥 = 𝑏 → ((𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏)))
20	19	rspcv 3631	. . . . . . 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 725	. . . . 5 ⊢ (((𝑏 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝑏) → (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏)))
23		sstr2 4015	. . . . 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 7937	. 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 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976  𝒫 cpw 4622  ∩ cint 4970  ran crn 5701  suc csuc 6397  ⟶wf 6569  ‘cfv 6573  ωcom 7903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-tr 5284  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fv 6581  df-om 7904

This theorem is referenced by:  isf32lem2  10423  isf32lem3  10424