Theorem isf32lem8

Description: Lemma for isfin3-2 . K sets are subsets of the base. (Contributed by Stefan O'Rear, 6-Nov-2014)

	Ref	Expression
Hypotheses	isf32lem.a	⊢ ( 𝜑 → 𝐹 : ω ⟶ 𝒫 𝐺 )
	isf32lem.b	⊢ ( 𝜑 → ∀ 𝑥 ∈ ω ( 𝐹 ‘ suc 𝑥 ) ⊆ ( 𝐹 ‘ 𝑥 ) )
	isf32lem.c	⊢ ( 𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹 )
	isf32lem.d	⊢ 𝑆 = { 𝑦 ∈ ω ∣ ( 𝐹 ‘ suc 𝑦 ) ⊊ ( 𝐹 ‘ 𝑦 ) }
	isf32lem.e	⊢ 𝐽 = ( 𝑢 ∈ ω ↦ ( ℩ 𝑣 ∈ 𝑆 ( 𝑣 ∩ 𝑆 ) ≈ 𝑢 ) )
	isf32lem.f	⊢ 𝐾 = ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ∘ 𝐽 )
Assertion	isf32lem8	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( 𝐾 ‘ 𝐴 ) ⊆ 𝐺 )

Proof

Step	Hyp	Ref	Expression
1		isf32lem.a	⊢ ( 𝜑 → 𝐹 : ω ⟶ 𝒫 𝐺 )
2		isf32lem.b	⊢ ( 𝜑 → ∀ 𝑥 ∈ ω ( 𝐹 ‘ suc 𝑥 ) ⊆ ( 𝐹 ‘ 𝑥 ) )
3		isf32lem.c	⊢ ( 𝜑 → ¬ ∩ ran 𝐹 ∈ ran 𝐹 )
4		isf32lem.d	⊢ 𝑆 = { 𝑦 ∈ ω ∣ ( 𝐹 ‘ suc 𝑦 ) ⊊ ( 𝐹 ‘ 𝑦 ) }
5		isf32lem.e	⊢ 𝐽 = ( 𝑢 ∈ ω ↦ ( ℩ 𝑣 ∈ 𝑆 ( 𝑣 ∩ 𝑆 ) ≈ 𝑢 ) )
6		isf32lem.f	⊢ 𝐾 = ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ∘ 𝐽 )
7	6	fveq1i	⊢ ( 𝐾 ‘ 𝐴 ) = ( ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ∘ 𝐽 ) ‘ 𝐴 )
8	4	ssrab3	⊢ 𝑆 ⊆ ω
9	1 2 3 4	isf32lem5	⊢ ( 𝜑 → ¬ 𝑆 ∈ Fin )
10	5	fin23lem22	⊢ ( ( 𝑆 ⊆ ω ∧ ¬ 𝑆 ∈ Fin ) → 𝐽 : ω –1-1-onto→ 𝑆 )
11	8 9 10	sylancr	⊢ ( 𝜑 → 𝐽 : ω –1-1-onto→ 𝑆 )
12		f1of	⊢ ( 𝐽 : ω –1-1-onto→ 𝑆 → 𝐽 : ω ⟶ 𝑆 )
13	11 12	syl	⊢ ( 𝜑 → 𝐽 : ω ⟶ 𝑆 )
14		fvco3	⊢ ( ( 𝐽 : ω ⟶ 𝑆 ∧ 𝐴 ∈ ω ) → ( ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ∘ 𝐽 ) ‘ 𝐴 ) = ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ‘ ( 𝐽 ‘ 𝐴 ) ) )
15	13 14	sylan	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ∘ 𝐽 ) ‘ 𝐴 ) = ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ‘ ( 𝐽 ‘ 𝐴 ) ) )
16	13	ffvelrnda	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( 𝐽 ‘ 𝐴 ) ∈ 𝑆 )
17		fveq2	⊢ ( 𝑤 = ( 𝐽 ‘ 𝐴 ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) )
18		suceq	⊢ ( 𝑤 = ( 𝐽 ‘ 𝐴 ) → suc 𝑤 = suc ( 𝐽 ‘ 𝐴 ) )
19	18	fveq2d	⊢ ( 𝑤 = ( 𝐽 ‘ 𝐴 ) → ( 𝐹 ‘ suc 𝑤 ) = ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) )
20	17 19	difeq12d	⊢ ( 𝑤 = ( 𝐽 ‘ 𝐴 ) → ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) )
21		eqid	⊢ ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) = ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) )
22		fvex	⊢ ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∈ V
23	22	difexi	⊢ ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) ∈ V
24	20 21 23	fvmpt	⊢ ( ( 𝐽 ‘ 𝐴 ) ∈ 𝑆 → ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ‘ ( 𝐽 ‘ 𝐴 ) ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) )
25	16 24	syl	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ‘ ( 𝐽 ‘ 𝐴 ) ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) )
26	15 25	eqtrd	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ∘ 𝐽 ) ‘ 𝐴 ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) )
27	7 26	syl5eq	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( 𝐾 ‘ 𝐴 ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) )
28	1	adantr	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → 𝐹 : ω ⟶ 𝒫 𝐺 )
29	8 16	sseldi	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( 𝐽 ‘ 𝐴 ) ∈ ω )
30	28 29	ffvelrnd	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∈ 𝒫 𝐺 )
31	30	elpwid	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ⊆ 𝐺 )
32	31	ssdifssd	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) ⊆ 𝐺 )
33	27 32	eqsstrd	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( 𝐾 ‘ 𝐴 ) ⊆ 𝐺 )