Theorem isf32lem6

Description: Lemma for isfin3-2 . Each K value is nonempty. (Contributed by Stefan O'Rear, 5-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	isf32lem6	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( 𝐾 ‘ 𝐴 ) ≠ ∅ )

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	9	adantr	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ¬ 𝑆 ∈ Fin )
17	8 16 10	sylancr	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → 𝐽 : ω –1-1-onto→ 𝑆 )
18	17 12	syl	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → 𝐽 : ω ⟶ 𝑆 )
19		ffvelrn	⊢ ( ( 𝐽 : ω ⟶ 𝑆 ∧ 𝐴 ∈ ω ) → ( 𝐽 ‘ 𝐴 ) ∈ 𝑆 )
20	18 19	sylancom	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( 𝐽 ‘ 𝐴 ) ∈ 𝑆 )
21		fveq2	⊢ ( 𝑤 = ( 𝐽 ‘ 𝐴 ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) )
22		suceq	⊢ ( 𝑤 = ( 𝐽 ‘ 𝐴 ) → suc 𝑤 = suc ( 𝐽 ‘ 𝐴 ) )
23	22	fveq2d	⊢ ( 𝑤 = ( 𝐽 ‘ 𝐴 ) → ( 𝐹 ‘ suc 𝑤 ) = ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) )
24	21 23	difeq12d	⊢ ( 𝑤 = ( 𝐽 ‘ 𝐴 ) → ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) )
25		eqid	⊢ ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) = ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) )
26		fvex	⊢ ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∈ V
27	26	difexi	⊢ ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) ∈ V
28	24 25 27	fvmpt	⊢ ( ( 𝐽 ‘ 𝐴 ) ∈ 𝑆 → ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ‘ ( 𝐽 ‘ 𝐴 ) ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) )
29	20 28	syl	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ‘ ( 𝐽 ‘ 𝐴 ) ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) )
30	15 29	eqtrd	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ∘ 𝐽 ) ‘ 𝐴 ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) )
31	7 30	eqtrid	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( 𝐾 ‘ 𝐴 ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) )
32		suceq	⊢ ( 𝑦 = ( 𝐽 ‘ 𝐴 ) → suc 𝑦 = suc ( 𝐽 ‘ 𝐴 ) )
33	32	fveq2d	⊢ ( 𝑦 = ( 𝐽 ‘ 𝐴 ) → ( 𝐹 ‘ suc 𝑦 ) = ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) )
34		fveq2	⊢ ( 𝑦 = ( 𝐽 ‘ 𝐴 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) )
35	33 34	psseq12d	⊢ ( 𝑦 = ( 𝐽 ‘ 𝐴 ) → ( ( 𝐹 ‘ suc 𝑦 ) ⊊ ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ⊊ ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ) )
36	35 4	elrab2	⊢ ( ( 𝐽 ‘ 𝐴 ) ∈ 𝑆 ↔ ( ( 𝐽 ‘ 𝐴 ) ∈ ω ∧ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ⊊ ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ) )
37	36	simprbi	⊢ ( ( 𝐽 ‘ 𝐴 ) ∈ 𝑆 → ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ⊊ ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) )
38	20 37	syl	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ⊊ ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) )
39		df-pss	⊢ ( ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ⊊ ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ↔ ( ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ⊆ ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∧ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ≠ ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ) )
40	38 39	sylib	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ⊆ ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∧ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ≠ ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ) )
41		pssdifn0	⊢ ( ( ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ⊆ ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∧ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ≠ ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ) → ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) ≠ ∅ )
42	40 41	syl	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) ≠ ∅ )
43	31 42	eqnetrd	⊢ ( ( 𝜑 ∧ 𝐴 ∈ ω ) → ( 𝐾 ‘ 𝐴 ) ≠ ∅ )