Theorem isf32lem1 9764

Description: Lemma for isfin3-2 9778. 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 6645	. . . . 5 ⊢ (𝑎 = 𝐵 → (𝐹‘𝑎) = (𝐹‘𝐵))
2	1	sseq1d 3946	. . . 4 ⊢ (𝑎 = 𝐵 → ((𝐹‘𝑎) ⊆ (𝐹‘𝐵) ↔ (𝐹‘𝐵) ⊆ (𝐹‘𝐵)))
3	2	imbi2d 344	. . 3 ⊢ (𝑎 = 𝐵 → ((𝜑 → (𝐹‘𝑎) ⊆ (𝐹‘𝐵)) ↔ (𝜑 → (𝐹‘𝐵) ⊆ (𝐹‘𝐵))))
4		fveq2 6645	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝐹‘𝑎) = (𝐹‘𝑏))
5	4	sseq1d 3946	. . . 4 ⊢ (𝑎 = 𝑏 → ((𝐹‘𝑎) ⊆ (𝐹‘𝐵) ↔ (𝐹‘𝑏) ⊆ (𝐹‘𝐵)))
6	5	imbi2d 344	. . 3 ⊢ (𝑎 = 𝑏 → ((𝜑 → (𝐹‘𝑎) ⊆ (𝐹‘𝐵)) ↔ (𝜑 → (𝐹‘𝑏) ⊆ (𝐹‘𝐵))))
7		fveq2 6645	. . . . 5 ⊢ (𝑎 = suc 𝑏 → (𝐹‘𝑎) = (𝐹‘suc 𝑏))
8	7	sseq1d 3946	. . . 4 ⊢ (𝑎 = suc 𝑏 → ((𝐹‘𝑎) ⊆ (𝐹‘𝐵) ↔ (𝐹‘suc 𝑏) ⊆ (𝐹‘𝐵)))
9	8	imbi2d 344	. . 3 ⊢ (𝑎 = suc 𝑏 → ((𝜑 → (𝐹‘𝑎) ⊆ (𝐹‘𝐵)) ↔ (𝜑 → (𝐹‘suc 𝑏) ⊆ (𝐹‘𝐵))))
10		fveq2 6645	. . . . 5 ⊢ (𝑎 = 𝐴 → (𝐹‘𝑎) = (𝐹‘𝐴))
11	10	sseq1d 3946	. . . 4 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑎) ⊆ (𝐹‘𝐵) ↔ (𝐹‘𝐴) ⊆ (𝐹‘𝐵)))
12	11	imbi2d 344	. . 3 ⊢ (𝑎 = 𝐴 → ((𝜑 → (𝐹‘𝑎) ⊆ (𝐹‘𝐵)) ↔ (𝜑 → (𝐹‘𝐴) ⊆ (𝐹‘𝐵))))
13		ssid 3937	. . . 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 6224	. . . . . . . . . 10 ⊢ (𝑥 = 𝑏 → suc 𝑥 = suc 𝑏)
17	16	fveq2d 6649	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (𝐹‘suc 𝑥) = (𝐹‘suc 𝑏))
18		fveq2 6645	. . . . . . . . 9 ⊢ (𝑥 = 𝑏 → (𝐹‘𝑥) = (𝐹‘𝑏))
19	17, 18	sseq12d 3948	. . . . . . . 8 ⊢ (𝑥 = 𝑏 → ((𝐹‘suc 𝑥) ⊆ (𝐹‘𝑥) ↔ (𝐹‘suc 𝑏) ⊆ (𝐹‘𝑏)))
20	19	rspcv 3566	. . . . . . 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 3922	. . . . 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 7590	. 2 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ 𝐵 ⊆ 𝐴) → (𝜑 → (𝐹‘𝐴) ⊆ (𝐹‘𝐵)))
27	26	impr 458	1 ⊢ (((𝐴 ∈ ω ∧ 𝐵 ∈ ω) ∧ (𝐵 ⊆ 𝐴 ∧ 𝜑)) → (𝐹‘𝐴) ⊆ (𝐹‘𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3106   ⊆ wss 3881  𝒫 cpw 4497  ∩ cint 4838  ran crn 5520  suc csuc 6161  ⟶wf 6320  ‘cfv 6324  ωcom 7560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fv 6332  df-om 7561

This theorem is referenced by:  isf32lem2  9765  isf32lem3  9766