Proof of Theorem isf34lem7
Step | Hyp | Ref
| Expression |
1 | | compss.a |
. . . . . . 7
⊢ 𝐹 = (𝑥 ∈ 𝒫 𝐴 ↦ (𝐴 ∖ 𝑥)) |
2 | 1 | isf34lem2 9195 |
. . . . . 6
⊢ (𝐴 ∈ FinIII →
𝐹:𝒫 𝐴⟶𝒫 𝐴) |
3 | 2 | adantr 481 |
. . . . 5
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → 𝐹:𝒫 𝐴⟶𝒫 𝐴) |
4 | 3 | 3adant3 1081 |
. . . 4
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐹:𝒫 𝐴⟶𝒫 𝐴) |
5 | | ffn 6045 |
. . . 4
⊢ (𝐹:𝒫 𝐴⟶𝒫 𝐴 → 𝐹 Fn 𝒫 𝐴) |
6 | 4, 5 | syl 17 |
. . 3
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐹 Fn 𝒫 𝐴) |
7 | | imassrn 5477 |
. . . 4
⊢ (𝐹 “ ran 𝐺) ⊆ ran 𝐹 |
8 | | frn 6053 |
. . . . . 6
⊢ (𝐹:𝒫 𝐴⟶𝒫 𝐴 → ran 𝐹 ⊆ 𝒫 𝐴) |
9 | 3, 8 | syl 17 |
. . . . 5
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → ran 𝐹 ⊆ 𝒫 𝐴) |
10 | 9 | 3adant3 1081 |
. . . 4
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ran 𝐹 ⊆ 𝒫 𝐴) |
11 | 7, 10 | syl5ss 3614 |
. . 3
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴) |
12 | | simp1 1061 |
. . . . 5
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → 𝐴 ∈ FinIII) |
13 | | fco 6058 |
. . . . . . 7
⊢ ((𝐹:𝒫 𝐴⟶𝒫 𝐴 ∧ 𝐺:ω⟶𝒫 𝐴) → (𝐹 ∘ 𝐺):ω⟶𝒫 𝐴) |
14 | 2, 13 | sylan 488 |
. . . . . 6
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → (𝐹 ∘ 𝐺):ω⟶𝒫 𝐴) |
15 | 14 | 3adant3 1081 |
. . . . 5
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹 ∘ 𝐺):ω⟶𝒫 𝐴) |
16 | | sscon 3744 |
. . . . . . . 8
⊢ ((𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦) → (𝐴 ∖ (𝐺‘suc 𝑦)) ⊆ (𝐴 ∖ (𝐺‘𝑦))) |
17 | | simpr 477 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → 𝐺:ω⟶𝒫 𝐴) |
18 | | peano2 7086 |
. . . . . . . . . . 11
⊢ (𝑦 ∈ ω → suc 𝑦 ∈
ω) |
19 | | fvco3 6275 |
. . . . . . . . . . 11
⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘suc 𝑦) = (𝐹‘(𝐺‘suc 𝑦))) |
20 | 17, 18, 19 | syl2an 494 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘suc 𝑦) = (𝐹‘(𝐺‘suc 𝑦))) |
21 | | simpll 790 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) ∧ 𝑦 ∈ ω) → 𝐴 ∈ FinIII) |
22 | | ffvelrn 6357 |
. . . . . . . . . . . . 13
⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ suc 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ∈ 𝒫 𝐴) |
23 | 17, 18, 22 | syl2an 494 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ∈ 𝒫 𝐴) |
24 | 23 | elpwid 4170 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘suc 𝑦) ⊆ 𝐴) |
25 | 1 | isf34lem1 9194 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ FinIII ∧
(𝐺‘suc 𝑦) ⊆ 𝐴) → (𝐹‘(𝐺‘suc 𝑦)) = (𝐴 ∖ (𝐺‘suc 𝑦))) |
26 | 21, 24, 25 | syl2anc 693 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) ∧ 𝑦 ∈ ω) → (𝐹‘(𝐺‘suc 𝑦)) = (𝐴 ∖ (𝐺‘suc 𝑦))) |
27 | 20, 26 | eqtrd 2656 |
. . . . . . . . 9
⊢ (((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘suc 𝑦) = (𝐴 ∖ (𝐺‘suc 𝑦))) |
28 | | fvco3 6275 |
. . . . . . . . . . 11
⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦))) |
29 | 28 | adantll 750 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐹‘(𝐺‘𝑦))) |
30 | | ffvelrn 6357 |
. . . . . . . . . . . . 13
⊢ ((𝐺:ω⟶𝒫 𝐴 ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ∈ 𝒫 𝐴) |
31 | 30 | adantll 750 |
. . . . . . . . . . . 12
⊢ (((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ∈ 𝒫 𝐴) |
32 | 31 | elpwid 4170 |
. . . . . . . . . . 11
⊢ (((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) ∧ 𝑦 ∈ ω) → (𝐺‘𝑦) ⊆ 𝐴) |
33 | 1 | isf34lem1 9194 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ FinIII ∧
(𝐺‘𝑦) ⊆ 𝐴) → (𝐹‘(𝐺‘𝑦)) = (𝐴 ∖ (𝐺‘𝑦))) |
34 | 21, 32, 33 | syl2anc 693 |
. . . . . . . . . 10
⊢ (((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) ∧ 𝑦 ∈ ω) → (𝐹‘(𝐺‘𝑦)) = (𝐴 ∖ (𝐺‘𝑦))) |
35 | 29, 34 | eqtrd 2656 |
. . . . . . . . 9
⊢ (((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) ∧ 𝑦 ∈ ω) → ((𝐹 ∘ 𝐺)‘𝑦) = (𝐴 ∖ (𝐺‘𝑦))) |
36 | 27, 35 | sseq12d 3634 |
. . . . . . . 8
⊢ (((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) ∧ 𝑦 ∈ ω) → (((𝐹 ∘ 𝐺)‘suc 𝑦) ⊆ ((𝐹 ∘ 𝐺)‘𝑦) ↔ (𝐴 ∖ (𝐺‘suc 𝑦)) ⊆ (𝐴 ∖ (𝐺‘𝑦)))) |
37 | 16, 36 | syl5ibr 236 |
. . . . . . 7
⊢ (((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) ∧ 𝑦 ∈ ω) → ((𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦) → ((𝐹 ∘ 𝐺)‘suc 𝑦) ⊆ ((𝐹 ∘ 𝐺)‘𝑦))) |
38 | 37 | ralimdva 2962 |
. . . . . 6
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → (∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦) → ∀𝑦 ∈ ω ((𝐹 ∘ 𝐺)‘suc 𝑦) ⊆ ((𝐹 ∘ 𝐺)‘𝑦))) |
39 | 38 | 3impia 1261 |
. . . . 5
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∀𝑦 ∈ ω ((𝐹 ∘ 𝐺)‘suc 𝑦) ⊆ ((𝐹 ∘ 𝐺)‘𝑦)) |
40 | | fin33i 9191 |
. . . . 5
⊢ ((𝐴 ∈ FinIII ∧
(𝐹 ∘ 𝐺):ω⟶𝒫 𝐴 ∧ ∀𝑦 ∈ ω ((𝐹 ∘ 𝐺)‘suc 𝑦) ⊆ ((𝐹 ∘ 𝐺)‘𝑦)) → ∩ ran
(𝐹 ∘ 𝐺) ∈ ran (𝐹 ∘ 𝐺)) |
41 | 12, 15, 39, 40 | syl3anc 1326 |
. . . 4
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∩ ran
(𝐹 ∘ 𝐺) ∈ ran (𝐹 ∘ 𝐺)) |
42 | | rnco2 5642 |
. . . . 5
⊢ ran
(𝐹 ∘ 𝐺) = (𝐹 “ ran 𝐺) |
43 | 42 | inteqi 4479 |
. . . 4
⊢ ∩ ran (𝐹 ∘ 𝐺) = ∩ (𝐹 “ ran 𝐺) |
44 | 41, 43, 42 | 3eltr3g 2717 |
. . 3
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∩ (𝐹 “ ran 𝐺) ∈ (𝐹 “ ran 𝐺)) |
45 | | fnfvima 6496 |
. . 3
⊢ ((𝐹 Fn 𝒫 𝐴 ∧ (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴 ∧ ∩ (𝐹 “ ran 𝐺) ∈ (𝐹 “ ran 𝐺)) → (𝐹‘∩ (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺))) |
46 | 6, 11, 44, 45 | syl3anc 1326 |
. 2
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → (𝐹‘∩ (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺))) |
47 | | simpl 473 |
. . . . . 6
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → 𝐴 ∈ FinIII) |
48 | 7, 9 | syl5ss 3614 |
. . . . . 6
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → (𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴) |
49 | | incom 3805 |
. . . . . . . . 9
⊢ (dom
𝐹 ∩ ran 𝐺) = (ran 𝐺 ∩ dom 𝐹) |
50 | | frn 6053 |
. . . . . . . . . . . 12
⊢ (𝐺:ω⟶𝒫 𝐴 → ran 𝐺 ⊆ 𝒫 𝐴) |
51 | 50 | adantl 482 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → ran 𝐺 ⊆ 𝒫 𝐴) |
52 | | fdm 6051 |
. . . . . . . . . . . 12
⊢ (𝐹:𝒫 𝐴⟶𝒫 𝐴 → dom 𝐹 = 𝒫 𝐴) |
53 | 3, 52 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → dom 𝐹 = 𝒫 𝐴) |
54 | 51, 53 | sseqtr4d 3642 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → ran 𝐺 ⊆ dom 𝐹) |
55 | | df-ss 3588 |
. . . . . . . . . 10
⊢ (ran
𝐺 ⊆ dom 𝐹 ↔ (ran 𝐺 ∩ dom 𝐹) = ran 𝐺) |
56 | 54, 55 | sylib 208 |
. . . . . . . . 9
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → (ran 𝐺 ∩ dom 𝐹) = ran 𝐺) |
57 | 49, 56 | syl5eq 2668 |
. . . . . . . 8
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → (dom 𝐹 ∩ ran 𝐺) = ran 𝐺) |
58 | | fdm 6051 |
. . . . . . . . . . 11
⊢ (𝐺:ω⟶𝒫 𝐴 → dom 𝐺 = ω) |
59 | 58 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → dom 𝐺 = ω) |
60 | | peano1 7085 |
. . . . . . . . . . 11
⊢ ∅
∈ ω |
61 | | ne0i 3921 |
. . . . . . . . . . 11
⊢ (∅
∈ ω → ω ≠ ∅) |
62 | 60, 61 | mp1i 13 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → ω ≠
∅) |
63 | 59, 62 | eqnetrd 2861 |
. . . . . . . . 9
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → dom 𝐺 ≠ ∅) |
64 | | dm0rn0 5342 |
. . . . . . . . . 10
⊢ (dom
𝐺 = ∅ ↔ ran
𝐺 =
∅) |
65 | 64 | necon3bii 2846 |
. . . . . . . . 9
⊢ (dom
𝐺 ≠ ∅ ↔ ran
𝐺 ≠
∅) |
66 | 63, 65 | sylib 208 |
. . . . . . . 8
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → ran 𝐺 ≠ ∅) |
67 | 57, 66 | eqnetrd 2861 |
. . . . . . 7
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → (dom 𝐹 ∩ ran 𝐺) ≠ ∅) |
68 | | imadisj 5484 |
. . . . . . . 8
⊢ ((𝐹 “ ran 𝐺) = ∅ ↔ (dom 𝐹 ∩ ran 𝐺) = ∅) |
69 | 68 | necon3bii 2846 |
. . . . . . 7
⊢ ((𝐹 “ ran 𝐺) ≠ ∅ ↔ (dom 𝐹 ∩ ran 𝐺) ≠ ∅) |
70 | 67, 69 | sylibr 224 |
. . . . . 6
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → (𝐹 “ ran 𝐺) ≠ ∅) |
71 | 1 | isf34lem5 9200 |
. . . . . 6
⊢ ((𝐴 ∈ FinIII ∧
((𝐹 “ ran 𝐺) ⊆ 𝒫 𝐴 ∧ (𝐹 “ ran 𝐺) ≠ ∅)) → (𝐹‘∩ (𝐹 “ ran 𝐺)) = ∪ (𝐹 “ (𝐹 “ ran 𝐺))) |
72 | 47, 48, 70, 71 | syl12anc 1324 |
. . . . 5
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → (𝐹‘∩ (𝐹
“ ran 𝐺)) = ∪ (𝐹
“ (𝐹 “ ran
𝐺))) |
73 | 1 | isf34lem3 9197 |
. . . . . . 7
⊢ ((𝐴 ∈ FinIII ∧
ran 𝐺 ⊆ 𝒫
𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺) |
74 | 47, 51, 73 | syl2anc 693 |
. . . . . 6
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → (𝐹 “ (𝐹 “ ran 𝐺)) = ran 𝐺) |
75 | 74 | unieqd 4446 |
. . . . 5
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → ∪ (𝐹
“ (𝐹 “ ran
𝐺)) = ∪ ran 𝐺) |
76 | 72, 75 | eqtrd 2656 |
. . . 4
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → (𝐹‘∩ (𝐹
“ ran 𝐺)) = ∪ ran 𝐺) |
77 | 76, 74 | eleq12d 2695 |
. . 3
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴) → ((𝐹‘∩ (𝐹
“ ran 𝐺)) ∈
(𝐹 “ (𝐹 “ ran 𝐺)) ↔ ∪ ran
𝐺 ∈ ran 𝐺)) |
78 | 77 | 3adant3 1081 |
. 2
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ((𝐹‘∩ (𝐹 “ ran 𝐺)) ∈ (𝐹 “ (𝐹 “ ran 𝐺)) ↔ ∪ ran
𝐺 ∈ ran 𝐺)) |
79 | 46, 78 | mpbid 222 |
1
⊢ ((𝐴 ∈ FinIII ∧
𝐺:ω⟶𝒫
𝐴 ∧ ∀𝑦 ∈ ω (𝐺‘𝑦) ⊆ (𝐺‘suc 𝑦)) → ∪ ran
𝐺 ∈ ran 𝐺) |