Metamath Proof Explorer

Theorem isf32lem7

Description: Lemma for isfin3-2 . Different K values are disjoint. (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 isf32lem7 ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝐾 ‘ 𝐴 ) ∩ ( 𝐾 ‘ 𝐵 ) ) = ∅ )

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 15 ad2ant2r ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ∘ 𝐽 ) ‘ 𝐴 ) = ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ‘ ( 𝐽 ‘ 𝐴 ) ) )
17 13 adantr ⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → 𝐽 : ω ⟶ 𝑆 )
18 simpl ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → 𝐴 ∈ ω )
19 ffvelrn ⊢ ( ( 𝐽 : ω ⟶ 𝑆 ∧ 𝐴 ∈ ω ) → ( 𝐽 ‘ 𝐴 ) ∈ 𝑆 )
20 17 18 19 syl2an ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ 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 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ‘ ( 𝐽 ‘ 𝐴 ) ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) )
30 16 29 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ∘ 𝐽 ) ‘ 𝐴 ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) )
31 7 30 eqtrid ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝐾 ‘ 𝐴 ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) )
32 6 fveq1i ⊢ ( 𝐾 ‘ 𝐵 ) = ( ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ∘ 𝐽 ) ‘ 𝐵 )
33 fvco3 ⊢ ( ( 𝐽 : ω ⟶ 𝑆 ∧ 𝐵 ∈ ω ) → ( ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ∘ 𝐽 ) ‘ 𝐵 ) = ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ‘ ( 𝐽 ‘ 𝐵 ) ) )
34 13 33 sylan ⊢ ( ( 𝜑 ∧ 𝐵 ∈ ω ) → ( ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ∘ 𝐽 ) ‘ 𝐵 ) = ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ‘ ( 𝐽 ‘ 𝐵 ) ) )
35 34 ad2ant2rl ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ∘ 𝐽 ) ‘ 𝐵 ) = ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ‘ ( 𝐽 ‘ 𝐵 ) ) )
36 simpr ⊢ ( ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) → 𝐵 ∈ ω )
37 ffvelrn ⊢ ( ( 𝐽 : ω ⟶ 𝑆 ∧ 𝐵 ∈ ω ) → ( 𝐽 ‘ 𝐵 ) ∈ 𝑆 )
38 17 36 37 syl2an ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝐽 ‘ 𝐵 ) ∈ 𝑆 )
39 fveq2 ⊢ ( 𝑤 = ( 𝐽 ‘ 𝐵 ) → ( 𝐹 ‘ 𝑤 ) = ( 𝐹 ‘ ( 𝐽 ‘ 𝐵 ) ) )
40 suceq ⊢ ( 𝑤 = ( 𝐽 ‘ 𝐵 ) → suc 𝑤 = suc ( 𝐽 ‘ 𝐵 ) )
41 40 fveq2d ⊢ ( 𝑤 = ( 𝐽 ‘ 𝐵 ) → ( 𝐹 ‘ suc 𝑤 ) = ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐵 ) ) )
42 39 41 difeq12d ⊢ ( 𝑤 = ( 𝐽 ‘ 𝐵 ) → ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐵 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐵 ) ) ) )
43 fvex ⊢ ( 𝐹 ‘ ( 𝐽 ‘ 𝐵 ) ) ∈ V
44 43 difexi ⊢ ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐵 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐵 ) ) ) ∈ V
45 42 25 44 fvmpt ⊢ ( ( 𝐽 ‘ 𝐵 ) ∈ 𝑆 → ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ‘ ( 𝐽 ‘ 𝐵 ) ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐵 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐵 ) ) ) )
46 38 45 syl ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ‘ ( 𝐽 ‘ 𝐵 ) ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐵 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐵 ) ) ) )
47 35 46 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( ( 𝑤 ∈ 𝑆 ↦ ( ( 𝐹 ‘ 𝑤 ) ∖ ( 𝐹 ‘ suc 𝑤 ) ) ) ∘ 𝐽 ) ‘ 𝐵 ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐵 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐵 ) ) ) )
48 32 47 eqtrid ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝐾 ‘ 𝐵 ) = ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐵 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐵 ) ) ) )
49 31 48 ineq12d ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝐾 ‘ 𝐴 ) ∩ ( 𝐾 ‘ 𝐵 ) ) = ( ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) ∩ ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐵 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐵 ) ) ) ) )
50 simpll ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → 𝜑 )
51 simplr ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → 𝐴 ≠ 𝐵 )
52 f1of1 ⊢ ( 𝐽 : ω –1-1-onto→ 𝑆 → 𝐽 : ω –1-1→ 𝑆 )
53 11 52 syl ⊢ ( 𝜑 → 𝐽 : ω –1-1→ 𝑆 )
54 53 adantr ⊢ ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) → 𝐽 : ω –1-1→ 𝑆 )
55 f1fveq ⊢ ( ( 𝐽 : ω –1-1→ 𝑆 ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝐽 ‘ 𝐴 ) = ( 𝐽 ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )
56 54 55 sylan ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝐽 ‘ 𝐴 ) = ( 𝐽 ‘ 𝐵 ) ↔ 𝐴 = 𝐵 ) )
57 56 biimpd ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝐽 ‘ 𝐴 ) = ( 𝐽 ‘ 𝐵 ) → 𝐴 = 𝐵 ) )
58 57 necon3d ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝐴 ≠ 𝐵 → ( 𝐽 ‘ 𝐴 ) ≠ ( 𝐽 ‘ 𝐵 ) ) )
59 51 58 mpd ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝐽 ‘ 𝐴 ) ≠ ( 𝐽 ‘ 𝐵 ) )
60 8 20 sselid ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝐽 ‘ 𝐴 ) ∈ ω )
61 8 38 sselid ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( 𝐽 ‘ 𝐵 ) ∈ ω )
62 1 2 3 isf32lem4 ⊢ ( ( ( 𝜑 ∧ ( 𝐽 ‘ 𝐴 ) ≠ ( 𝐽 ‘ 𝐵 ) ) ∧ ( ( 𝐽 ‘ 𝐴 ) ∈ ω ∧ ( 𝐽 ‘ 𝐵 ) ∈ ω ) ) → ( ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) ∩ ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐵 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐵 ) ) ) ) = ∅ )
63 50 59 60 61 62 syl22anc ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐴 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐴 ) ) ) ∩ ( ( 𝐹 ‘ ( 𝐽 ‘ 𝐵 ) ) ∖ ( 𝐹 ‘ suc ( 𝐽 ‘ 𝐵 ) ) ) ) = ∅ )
64 49 63 eqtrd ⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ 𝐵 ) ∧ ( 𝐴 ∈ ω ∧ 𝐵 ∈ ω ) ) → ( ( 𝐾 ‘ 𝐴 ) ∩ ( 𝐾 ‘ 𝐵 ) ) = ∅ )