Step | Hyp | Ref
| Expression |
1 | | tfrfun 6270 |
. . . 4
⊢ Fun
recs(𝐺) |
2 | | tfri1dALT.1 |
. . . . 5
⊢ 𝐹 = recs(𝐺) |
3 | 2 | funeqi 5194 |
. . . 4
⊢ (Fun
𝐹 ↔ Fun recs(𝐺)) |
4 | 1, 3 | mpbir 145 |
. . 3
⊢ Fun 𝐹 |
5 | 4 | a1i 9 |
. 2
⊢ (𝜑 → Fun 𝐹) |
6 | | eqid 2157 |
. . . . . 6
⊢ {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))} = {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐 ∈ 𝑏 (𝑎‘𝑐) = (𝐺‘(𝑎 ↾ 𝑐)))} |
7 | 6 | tfrlem8 6268 |
. . . . 5
⊢ Ord dom
recs(𝐺) |
8 | 2 | dmeqi 4790 |
. . . . . 6
⊢ dom 𝐹 = dom recs(𝐺) |
9 | | ordeq 4335 |
. . . . . 6
⊢ (dom
𝐹 = dom recs(𝐺) → (Ord dom 𝐹 ↔ Ord dom recs(𝐺))) |
10 | 8, 9 | ax-mp 5 |
. . . . 5
⊢ (Ord dom
𝐹 ↔ Ord dom recs(𝐺)) |
11 | 7, 10 | mpbir 145 |
. . . 4
⊢ Ord dom
𝐹 |
12 | | ordsson 4454 |
. . . 4
⊢ (Ord dom
𝐹 → dom 𝐹 ⊆ On) |
13 | 11, 12 | mp1i 10 |
. . 3
⊢ (𝜑 → dom 𝐹 ⊆ On) |
14 | | tfri1dALT.2 |
. . . . . . . . . 10
⊢ (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V)) |
15 | | simpl 108 |
. . . . . . . . . . 11
⊢ ((Fun
𝐺 ∧ (𝐺‘𝑥) ∈ V) → Fun 𝐺) |
16 | 15 | alimi 1435 |
. . . . . . . . . 10
⊢
(∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) → ∀𝑥Fun 𝐺) |
17 | 14, 16 | syl 14 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑥Fun 𝐺) |
18 | 17 | 19.21bi 1538 |
. . . . . . . 8
⊢ (𝜑 → Fun 𝐺) |
19 | 18 | adantr 274 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ On) → Fun 𝐺) |
20 | | ordon 4448 |
. . . . . . . 8
⊢ Ord
On |
21 | 20 | a1i 9 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ On) → Ord On) |
22 | | simpr 109 |
. . . . . . . . . . 11
⊢ ((Fun
𝐺 ∧ (𝐺‘𝑥) ∈ V) → (𝐺‘𝑥) ∈ V) |
23 | 22 | alimi 1435 |
. . . . . . . . . 10
⊢
(∀𝑥(Fun 𝐺 ∧ (𝐺‘𝑥) ∈ V) → ∀𝑥(𝐺‘𝑥) ∈ V) |
24 | | fveq2 5471 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑓 → (𝐺‘𝑥) = (𝐺‘𝑓)) |
25 | 24 | eleq1d 2226 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑓 → ((𝐺‘𝑥) ∈ V ↔ (𝐺‘𝑓) ∈ V)) |
26 | 25 | spv 1840 |
. . . . . . . . . 10
⊢
(∀𝑥(𝐺‘𝑥) ∈ V → (𝐺‘𝑓) ∈ V) |
27 | 14, 23, 26 | 3syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (𝐺‘𝑓) ∈ V) |
28 | 27 | adantr 274 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑧 ∈ On) → (𝐺‘𝑓) ∈ V) |
29 | 28 | 3ad2ant1 1003 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓 Fn 𝑦) → (𝐺‘𝑓) ∈ V) |
30 | | suceloni 4463 |
. . . . . . . . 9
⊢ (𝑦 ∈ On → suc 𝑦 ∈ On) |
31 | | unon 4473 |
. . . . . . . . 9
⊢ ∪ On = On |
32 | 30, 31 | eleq2s 2252 |
. . . . . . . 8
⊢ (𝑦 ∈ ∪ On → suc 𝑦 ∈ On) |
33 | 32 | adantl 275 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑧 ∈ On) ∧ 𝑦 ∈ ∪ On)
→ suc 𝑦 ∈
On) |
34 | | suceloni 4463 |
. . . . . . . 8
⊢ (𝑧 ∈ On → suc 𝑧 ∈ On) |
35 | 34 | adantl 275 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ On) → suc 𝑧 ∈ On) |
36 | 2, 19, 21, 29, 33, 35 | tfr1on 6300 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ On) → suc 𝑧 ⊆ dom 𝐹) |
37 | | vex 2715 |
. . . . . . 7
⊢ 𝑧 ∈ V |
38 | 37 | sucid 4380 |
. . . . . 6
⊢ 𝑧 ∈ suc 𝑧 |
39 | | ssel2 3123 |
. . . . . 6
⊢ ((suc
𝑧 ⊆ dom 𝐹 ∧ 𝑧 ∈ suc 𝑧) → 𝑧 ∈ dom 𝐹) |
40 | 36, 38, 39 | sylancl 410 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ On) → 𝑧 ∈ dom 𝐹) |
41 | 40 | ex 114 |
. . . 4
⊢ (𝜑 → (𝑧 ∈ On → 𝑧 ∈ dom 𝐹)) |
42 | 41 | ssrdv 3134 |
. . 3
⊢ (𝜑 → On ⊆ dom 𝐹) |
43 | 13, 42 | eqssd 3145 |
. 2
⊢ (𝜑 → dom 𝐹 = On) |
44 | | df-fn 5176 |
. 2
⊢ (𝐹 Fn On ↔ (Fun 𝐹 ∧ dom 𝐹 = On)) |
45 | 5, 43, 44 | sylanbrc 414 |
1
⊢ (𝜑 → 𝐹 Fn On) |