Proof of Theorem bnj945
Step | Hyp | Ref
| Expression |
1 | | fndm 6481 |
. . . . . . 7
⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛) |
2 | 1 | ad2antll 729 |
. . . . . 6
⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → dom 𝑓 = 𝑛) |
3 | 2 | eleq2d 2823 |
. . . . 5
⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → (𝐴 ∈ dom 𝑓 ↔ 𝐴 ∈ 𝑛)) |
4 | 3 | pm5.32i 578 |
. . . 4
⊢ (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ dom 𝑓) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ 𝑛)) |
5 | | bnj945.1 |
. . . . . . . . 9
⊢ 𝐺 = (𝑓 ∪ {〈𝑛, 𝐶〉}) |
6 | 5 | bnj941 32465 |
. . . . . . . 8
⊢ (𝐶 ∈ V → ((𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) → 𝐺 Fn 𝑝)) |
7 | 6 | imp 410 |
. . . . . . 7
⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → 𝐺 Fn 𝑝) |
8 | 7 | bnj930 32462 |
. . . . . 6
⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → Fun 𝐺) |
9 | 5 | bnj931 32463 |
. . . . . 6
⊢ 𝑓 ⊆ 𝐺 |
10 | 8, 9 | jctir 524 |
. . . . 5
⊢ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) → (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺)) |
11 | 10 | anim1i 618 |
. . . 4
⊢ (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ dom 𝑓) → ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ∧ 𝐴 ∈ dom 𝑓)) |
12 | 4, 11 | sylbir 238 |
. . 3
⊢ (((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ 𝑛) → ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ∧ 𝐴 ∈ dom 𝑓)) |
13 | | df-bnj17 32378 |
. . . 4
⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ∧ 𝐴 ∈ 𝑛)) |
14 | | 3ancomb 1101 |
. . . . . 6
⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) |
15 | | 3anass 1097 |
. . . . . 6
⊢ ((𝐶 ∈ V ∧ 𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛))) |
16 | 14, 15 | bitri 278 |
. . . . 5
⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ↔ (𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛))) |
17 | 16 | anbi1i 627 |
. . . 4
⊢ (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ∧ 𝐴 ∈ 𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ 𝑛)) |
18 | 13, 17 | bitri 278 |
. . 3
⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) ↔ ((𝐶 ∈ V ∧ (𝑝 = suc 𝑛 ∧ 𝑓 Fn 𝑛)) ∧ 𝐴 ∈ 𝑛)) |
19 | | df-3an 1091 |
. . 3
⊢ ((Fun
𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝐴 ∈ dom 𝑓) ↔ ((Fun 𝐺 ∧ 𝑓 ⊆ 𝐺) ∧ 𝐴 ∈ dom 𝑓)) |
20 | 12, 18, 19 | 3imtr4i 295 |
. 2
⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) → (Fun 𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝐴 ∈ dom 𝑓)) |
21 | | funssfv 6738 |
. 2
⊢ ((Fun
𝐺 ∧ 𝑓 ⊆ 𝐺 ∧ 𝐴 ∈ dom 𝑓) → (𝐺‘𝐴) = (𝑓‘𝐴)) |
22 | 20, 21 | syl 17 |
1
⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝐴 ∈ 𝑛) → (𝐺‘𝐴) = (𝑓‘𝐴)) |