Proof of Theorem bnj953
Step | Hyp | Ref
| Expression |
1 | | bnj312 32261 |
. . 3
⊢ (((𝐺‘𝑖) = (𝑓‘𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓) ↔ ((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝐺‘𝑖) = (𝑓‘𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓)) |
2 | | bnj251 32251 |
. . 3
⊢ (((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝐺‘𝑖) = (𝑓‘𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓) ↔ ((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ ((𝐺‘𝑖) = (𝑓‘𝑖) ∧ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓)))) |
3 | 1, 2 | bitri 278 |
. 2
⊢ (((𝐺‘𝑖) = (𝑓‘𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓) ↔ ((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ ((𝐺‘𝑖) = (𝑓‘𝑖) ∧ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓)))) |
4 | | bnj953.1 |
. . . . . 6
⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
5 | 4 | bnj115 32274 |
. . . . 5
⊢ (𝜓 ↔ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
6 | | sp 2184 |
. . . . . 6
⊢
(∀𝑖((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) → ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) |
7 | 6 | impcom 411 |
. . . . 5
⊢ (((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ ∀𝑖((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
8 | 5, 7 | sylan2b 597 |
. . . 4
⊢ (((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
9 | | bnj953.2 |
. . . . 5
⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖)) |
10 | 9 | bnj956 32327 |
. . . 4
⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∪
𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪
𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
11 | | eqtr3 2760 |
. . . 4
⊢ (((𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) ∧ ∪
𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪
𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
12 | 8, 10, 11 | syl2anr 600 |
. . 3
⊢ (((𝐺‘𝑖) = (𝑓‘𝑖) ∧ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓)) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
13 | | eqtr 2758 |
. . 3
⊢ (((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
14 | 12, 13 | sylan2 596 |
. 2
⊢ (((𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ ((𝐺‘𝑖) = (𝑓‘𝑖) ∧ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓))) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) |
15 | 3, 14 | sylbi 220 |
1
⊢ (((𝐺‘𝑖) = (𝑓‘𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)) |