Proof of Theorem bnj1097
Step | Hyp | Ref
| Expression |
1 | | bnj1097.3 |
. . . . . . . 8
⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) |
2 | | bnj1097.1 |
. . . . . . . . 9
⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
3 | 2 | biimpi 205 |
. . . . . . . 8
⊢ (𝜑 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
4 | 1, 3 | bnj771 30088 |
. . . . . . 7
⊢ (𝜒 → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
5 | 4 | 3ad2ant3 1077 |
. . . . . 6
⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
6 | 5 | adantr 480 |
. . . . 5
⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) |
7 | | bnj1097.5 |
. . . . . . . 8
⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)) |
8 | 7 | simp3bi 1071 |
. . . . . . 7
⊢ (𝜏 → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) |
9 | 8 | 3ad2ant2 1076 |
. . . . . 6
⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) |
10 | 9 | adantr 480 |
. . . . 5
⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) |
11 | 6, 10 | jca 553 |
. . . 4
⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)) |
12 | 11 | anim2i 591 |
. . 3
⊢ ((𝑖 = ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑖 = ∅ ∧ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))) |
13 | | 3anass 1035 |
. . 3
⊢ ((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) ↔ (𝑖 = ∅ ∧ ((𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))) |
14 | 12, 13 | sylibr 223 |
. 2
⊢ ((𝑖 = ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵)) |
15 | | fveq2 6103 |
. . . . . . 7
⊢ (𝑖 = ∅ → (𝑓‘𝑖) = (𝑓‘∅)) |
16 | 15 | eqeq1d 2612 |
. . . . . 6
⊢ (𝑖 = ∅ → ((𝑓‘𝑖) = pred(𝑋, 𝐴, 𝑅) ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))) |
17 | 16 | biimpar 501 |
. . . . 5
⊢ ((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝑓‘𝑖) = pred(𝑋, 𝐴, 𝑅)) |
18 | 17 | adantr 480 |
. . . 4
⊢ (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓‘𝑖) = pred(𝑋, 𝐴, 𝑅)) |
19 | | simpr 476 |
. . . 4
⊢ (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) |
20 | 18, 19 | eqsstrd 3602 |
. . 3
⊢ (((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓‘𝑖) ⊆ 𝐵) |
21 | 20 | 3impa 1251 |
. 2
⊢ ((𝑖 = ∅ ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵) → (𝑓‘𝑖) ⊆ 𝐵) |
22 | 14, 21 | syl 17 |
1
⊢ ((𝑖 = ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓‘𝑖) ⊆ 𝐵) |