Proof of Theorem tfrlemibfn
Step | Hyp | Ref
| Expression |
1 | | tfrlemisucfn.1 |
. . . . . 6
⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))} |
2 | | tfrlemisucfn.2 |
. . . . . 6
⊢ (𝜑 → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V)) |
3 | | tfrlemi1.3 |
. . . . . 6
⊢ 𝐵 = {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))} |
4 | | tfrlemi1.4 |
. . . . . 6
⊢ (𝜑 → 𝑥 ∈ On) |
5 | | tfrlemi1.5 |
. . . . . 6
⊢ (𝜑 → ∀𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))) |
6 | 1, 2, 3, 4, 5 | tfrlemibacc 6302 |
. . . . 5
⊢ (𝜑 → 𝐵 ⊆ 𝐴) |
7 | 6 | unissd 3818 |
. . . 4
⊢ (𝜑 → ∪ 𝐵
⊆ ∪ 𝐴) |
8 | 1 | recsfval 6291 |
. . . 4
⊢
recs(𝐹) = ∪ 𝐴 |
9 | 7, 8 | sseqtrrdi 3196 |
. . 3
⊢ (𝜑 → ∪ 𝐵
⊆ recs(𝐹)) |
10 | 1 | tfrlem7 6293 |
. . 3
⊢ Fun
recs(𝐹) |
11 | | funss 5215 |
. . 3
⊢ (∪ 𝐵
⊆ recs(𝐹) → (Fun
recs(𝐹) → Fun ∪ 𝐵)) |
12 | 9, 10, 11 | mpisyl 1439 |
. 2
⊢ (𝜑 → Fun ∪ 𝐵) |
13 | | simpr3 1000 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉})) |
14 | 2 | ad2antrr 485 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → ∀𝑥(Fun 𝐹 ∧ (𝐹‘𝑥) ∈ V)) |
15 | 4 | ad2antrr 485 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → 𝑥 ∈ On) |
16 | | simplr 525 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → 𝑧 ∈ 𝑥) |
17 | | onelon 4367 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ 𝑥) → 𝑧 ∈ On) |
18 | 15, 16, 17 | syl2anc 409 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → 𝑧 ∈ On) |
19 | | simpr1 998 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → 𝑔 Fn 𝑧) |
20 | | simpr2 999 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → 𝑔 ∈ 𝐴) |
21 | 1, 14, 18, 19, 20 | tfrlemisucfn 6300 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) Fn suc 𝑧) |
22 | | dffn2 5347 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) Fn suc 𝑧 ↔ (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}):suc 𝑧⟶V) |
23 | 21, 22 | sylib 121 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}):suc 𝑧⟶V) |
24 | | fssxp 5363 |
. . . . . . . . . . . . . . 15
⊢ ((𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}):suc 𝑧⟶V → (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) ⊆ (suc 𝑧 × V)) |
25 | 23, 24 | syl 14 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) ⊆ (suc 𝑧 × V)) |
26 | | eloni 4358 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 ∈ On → Ord 𝑥) |
27 | 15, 26 | syl 14 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → Ord 𝑥) |
28 | | ordsucss 4486 |
. . . . . . . . . . . . . . . 16
⊢ (Ord
𝑥 → (𝑧 ∈ 𝑥 → suc 𝑧 ⊆ 𝑥)) |
29 | 27, 16, 28 | sylc 62 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → suc 𝑧 ⊆ 𝑥) |
30 | | xpss1 4719 |
. . . . . . . . . . . . . . 15
⊢ (suc
𝑧 ⊆ 𝑥 → (suc 𝑧 × V) ⊆ (𝑥 × V)) |
31 | 29, 30 | syl 14 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → (suc 𝑧 × V) ⊆ (𝑥 × V)) |
32 | 25, 31 | sstrd 3157 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) ⊆ (𝑥 × V)) |
33 | | vex 2733 |
. . . . . . . . . . . . . . . 16
⊢ 𝑔 ∈ V |
34 | | vex 2733 |
. . . . . . . . . . . . . . . . . 18
⊢ 𝑧 ∈ V |
35 | 2 | tfrlem3-2d 6288 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → (Fun 𝐹 ∧ (𝐹‘𝑔) ∈ V)) |
36 | 35 | simprd 113 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐹‘𝑔) ∈ V) |
37 | | opexg 4211 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑧 ∈ V ∧ (𝐹‘𝑔) ∈ V) → 〈𝑧, (𝐹‘𝑔)〉 ∈ V) |
38 | 34, 36, 37 | sylancr 412 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 〈𝑧, (𝐹‘𝑔)〉 ∈ V) |
39 | | snexg 4168 |
. . . . . . . . . . . . . . . . 17
⊢
(〈𝑧, (𝐹‘𝑔)〉 ∈ V → {〈𝑧, (𝐹‘𝑔)〉} ∈ V) |
40 | 38, 39 | syl 14 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → {〈𝑧, (𝐹‘𝑔)〉} ∈ V) |
41 | | unexg 4426 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑔 ∈ V ∧ {〈𝑧, (𝐹‘𝑔)〉} ∈ V) → (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) ∈ V) |
42 | 33, 40, 41 | sylancr 412 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) ∈ V) |
43 | | elpwg 3572 |
. . . . . . . . . . . . . . 15
⊢ ((𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) ∈ V → ((𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) ∈ 𝒫 (𝑥 × V) ↔ (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) ⊆ (𝑥 × V))) |
44 | 42, 43 | syl 14 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) ∈ 𝒫 (𝑥 × V) ↔ (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) ⊆ (𝑥 × V))) |
45 | 44 | ad2antrr 485 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → ((𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) ∈ 𝒫 (𝑥 × V) ↔ (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) ⊆ (𝑥 × V))) |
46 | 32, 45 | mpbird 166 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}) ∈ 𝒫 (𝑥 × V)) |
47 | 13, 46 | eqeltrd 2247 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑧 ∈ 𝑥) ∧ (𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))) → ℎ ∈ 𝒫 (𝑥 × V)) |
48 | 47 | ex 114 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑥) → ((𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉})) → ℎ ∈ 𝒫 (𝑥 × V))) |
49 | 48 | exlimdv 1812 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑧 ∈ 𝑥) → (∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉})) → ℎ ∈ 𝒫 (𝑥 × V))) |
50 | 49 | rexlimdva 2587 |
. . . . . . . 8
⊢ (𝜑 → (∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉})) → ℎ ∈ 𝒫 (𝑥 × V))) |
51 | 50 | abssdv 3221 |
. . . . . . 7
⊢ (𝜑 → {ℎ ∣ ∃𝑧 ∈ 𝑥 ∃𝑔(𝑔 Fn 𝑧 ∧ 𝑔 ∈ 𝐴 ∧ ℎ = (𝑔 ∪ {〈𝑧, (𝐹‘𝑔)〉}))} ⊆ 𝒫 (𝑥 × V)) |
52 | 3, 51 | eqsstrid 3193 |
. . . . . 6
⊢ (𝜑 → 𝐵 ⊆ 𝒫 (𝑥 × V)) |
53 | | sspwuni 3955 |
. . . . . 6
⊢ (𝐵 ⊆ 𝒫 (𝑥 × V) ↔ ∪ 𝐵
⊆ (𝑥 ×
V)) |
54 | 52, 53 | sylib 121 |
. . . . 5
⊢ (𝜑 → ∪ 𝐵
⊆ (𝑥 ×
V)) |
55 | | dmss 4808 |
. . . . 5
⊢ (∪ 𝐵
⊆ (𝑥 × V)
→ dom ∪ 𝐵 ⊆ dom (𝑥 × V)) |
56 | 54, 55 | syl 14 |
. . . 4
⊢ (𝜑 → dom ∪ 𝐵
⊆ dom (𝑥 ×
V)) |
57 | | dmxpss 5039 |
. . . 4
⊢ dom
(𝑥 × V) ⊆ 𝑥 |
58 | 56, 57 | sstrdi 3159 |
. . 3
⊢ (𝜑 → dom ∪ 𝐵
⊆ 𝑥) |
59 | 1, 2, 3, 4, 5 | tfrlemibxssdm 6303 |
. . 3
⊢ (𝜑 → 𝑥 ⊆ dom ∪
𝐵) |
60 | 58, 59 | eqssd 3164 |
. 2
⊢ (𝜑 → dom ∪ 𝐵 =
𝑥) |
61 | | df-fn 5199 |
. 2
⊢ (∪ 𝐵 Fn
𝑥 ↔ (Fun ∪ 𝐵
∧ dom ∪ 𝐵 = 𝑥)) |
62 | 12, 60, 61 | sylanbrc 415 |
1
⊢ (𝜑 → ∪ 𝐵 Fn
𝑥) |