Step | Hyp | Ref
| Expression |
1 | | fnprb.a |
. . . . . . . 8
⊢ 𝐴 ∈ V |
2 | | fnprb.b |
. . . . . . . 8
⊢ 𝐵 ∈ V |
3 | 1, 2 | fnprb 7159 |
. . . . . . 7
⊢ (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}) |
4 | | tpidm23 4719 |
. . . . . . . . 9
⊢ {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵} |
5 | 4 | eqcomi 2746 |
. . . . . . . 8
⊢ {𝐴, 𝐵} = {𝐴, 𝐵, 𝐵} |
6 | 5 | fneq2i 6601 |
. . . . . . 7
⊢ (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 Fn {𝐴, 𝐵, 𝐵}) |
7 | | tpidm23 4719 |
. . . . . . . . 9
⊢
{⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} |
8 | 7 | eqcomi 2746 |
. . . . . . . 8
⊢
{⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} |
9 | 8 | eqeq2i 2750 |
. . . . . . 7
⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}) |
10 | 3, 6, 9 | 3bitr3i 301 |
. . . . . 6
⊢ (𝐹 Fn {𝐴, 𝐵, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}) |
11 | 10 | a1i 11 |
. . . . 5
⊢ (𝐵 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) |
12 | | tpeq3 4706 |
. . . . . 6
⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐵} = {𝐴, 𝐵, 𝐶}) |
13 | 12 | fneq2d 6597 |
. . . . 5
⊢ (𝐵 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐵} ↔ 𝐹 Fn {𝐴, 𝐵, 𝐶})) |
14 | | id 22 |
. . . . . . . 8
⊢ (𝐵 = 𝐶 → 𝐵 = 𝐶) |
15 | | fveq2 6843 |
. . . . . . . 8
⊢ (𝐵 = 𝐶 → (𝐹‘𝐵) = (𝐹‘𝐶)) |
16 | 14, 15 | opeq12d 4839 |
. . . . . . 7
⊢ (𝐵 = 𝐶 → ⟨𝐵, (𝐹‘𝐵)⟩ = ⟨𝐶, (𝐹‘𝐶)⟩) |
17 | 16 | tpeq3d 4709 |
. . . . . 6
⊢ (𝐵 = 𝐶 → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) |
18 | 17 | eqeq2d 2748 |
. . . . 5
⊢ (𝐵 = 𝐶 → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})) |
19 | 11, 13, 18 | 3bitr3d 309 |
. . . 4
⊢ (𝐵 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})) |
20 | 19 | a1d 25 |
. . 3
⊢ (𝐵 = 𝐶 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}))) |
21 | | fndm 6606 |
. . . . . . . 8
⊢ (𝐹 Fn {𝐴, 𝐵, 𝐶} → dom 𝐹 = {𝐴, 𝐵, 𝐶}) |
22 | | fvex 6856 |
. . . . . . . . 9
⊢ (𝐹‘𝐴) ∈ V |
23 | | fvex 6856 |
. . . . . . . . 9
⊢ (𝐹‘𝐵) ∈ V |
24 | | fvex 6856 |
. . . . . . . . 9
⊢ (𝐹‘𝐶) ∈ V |
25 | 22, 23, 24 | dmtpop 6171 |
. . . . . . . 8
⊢ dom
{⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} = {𝐴, 𝐵, 𝐶} |
26 | 21, 25 | eqtr4di 2795 |
. . . . . . 7
⊢ (𝐹 Fn {𝐴, 𝐵, 𝐶} → dom 𝐹 = dom {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) |
27 | 26 | adantl 483 |
. . . . . 6
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → dom 𝐹 = dom {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) |
28 | 21 | adantl 483 |
. . . . . . . . 9
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → dom 𝐹 = {𝐴, 𝐵, 𝐶}) |
29 | 28 | eleq2d 2824 |
. . . . . . . 8
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ {𝐴, 𝐵, 𝐶})) |
30 | | vex 3450 |
. . . . . . . . . 10
⊢ 𝑥 ∈ V |
31 | 30 | eltp 4650 |
. . . . . . . . 9
⊢ (𝑥 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶)) |
32 | 1, 22 | fvtp1 7145 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐴) = (𝐹‘𝐴)) |
33 | 32 | ad2antrr 725 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐴) = (𝐹‘𝐴)) |
34 | 33 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝐹‘𝐴) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐴)) |
35 | | fveq2 6843 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) |
36 | | fveq2 6843 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝐴 → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐴)) |
37 | 35, 36 | eqeq12d 2753 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥) ↔ (𝐹‘𝐴) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐴))) |
38 | 34, 37 | syl5ibrcom 247 |
. . . . . . . . . 10
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 = 𝐴 → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥))) |
39 | 2, 23 | fvtp2 7146 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶) → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐵) = (𝐹‘𝐵)) |
40 | 39 | ad4ant13 750 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐵) = (𝐹‘𝐵)) |
41 | 40 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝐹‘𝐵) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐵)) |
42 | | fveq2 6843 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵)) |
43 | | fveq2 6843 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝐵 → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐵)) |
44 | 42, 43 | eqeq12d 2753 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥) ↔ (𝐹‘𝐵) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐵))) |
45 | 41, 44 | syl5ibrcom 247 |
. . . . . . . . . 10
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 = 𝐵 → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥))) |
46 | | fntpb.c |
. . . . . . . . . . . . . 14
⊢ 𝐶 ∈ V |
47 | 46, 24 | fvtp3 7147 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐶) = (𝐹‘𝐶)) |
48 | 47 | ad4ant23 752 |
. . . . . . . . . . . 12
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐶) = (𝐹‘𝐶)) |
49 | 48 | eqcomd 2743 |
. . . . . . . . . . 11
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝐹‘𝐶) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐶)) |
50 | | fveq2 6843 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝐶 → (𝐹‘𝑥) = (𝐹‘𝐶)) |
51 | | fveq2 6843 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝐶 → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐶)) |
52 | 50, 51 | eqeq12d 2753 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝐶 → ((𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥) ↔ (𝐹‘𝐶) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝐶))) |
53 | 49, 52 | syl5ibrcom 247 |
. . . . . . . . . 10
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 = 𝐶 → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥))) |
54 | 38, 45, 53 | 3jaod 1429 |
. . . . . . . . 9
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵 ∨ 𝑥 = 𝐶) → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥))) |
55 | 31, 54 | biimtrid 241 |
. . . . . . . 8
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 ∈ {𝐴, 𝐵, 𝐶} → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥))) |
56 | 29, 55 | sylbid 239 |
. . . . . . 7
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥))) |
57 | 56 | ralrimiv 3143 |
. . . . . 6
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥)) |
58 | | fnfun 6603 |
. . . . . . 7
⊢ (𝐹 Fn {𝐴, 𝐵, 𝐶} → Fun 𝐹) |
59 | 1, 2, 46, 22, 23, 24 | funtp 6559 |
. . . . . . . 8
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → Fun {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) |
60 | 59 | 3expa 1119 |
. . . . . . 7
⊢ (((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → Fun {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) |
61 | | eqfunfv 6988 |
. . . . . . 7
⊢ ((Fun
𝐹 ∧ Fun {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} ↔ (dom 𝐹 = dom {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥)))) |
62 | 58, 60, 61 | syl2anr 598 |
. . . . . 6
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} ↔ (dom 𝐹 = dom {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}‘𝑥)))) |
63 | 27, 57, 62 | mpbir2and 712 |
. . . . 5
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 Fn {𝐴, 𝐵, 𝐶}) → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) |
64 | 1, 2, 46, 22, 23, 24 | fntp 6563 |
. . . . . . 7
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ∧ 𝐵 ≠ 𝐶) → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} Fn {𝐴, 𝐵, 𝐶}) |
65 | 64 | 3expa 1119 |
. . . . . 6
⊢ (((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} Fn {𝐴, 𝐵, 𝐶}) |
66 | | fneq1 6594 |
. . . . . . 7
⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} Fn {𝐴, 𝐵, 𝐶})) |
67 | 66 | biimprd 248 |
. . . . . 6
⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} Fn {𝐴, 𝐵, 𝐶} → 𝐹 Fn {𝐴, 𝐵, 𝐶})) |
68 | 65, 67 | mpan9 508 |
. . . . 5
⊢ ((((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) → 𝐹 Fn {𝐴, 𝐵, 𝐶}) |
69 | 63, 68 | impbida 800 |
. . . 4
⊢ (((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) ∧ 𝐵 ≠ 𝐶) → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})) |
70 | 69 | expcom 415 |
. . 3
⊢ (𝐵 ≠ 𝐶 → ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}))) |
71 | 20, 70 | pm2.61ine 3029 |
. 2
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶) → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})) |
72 | 1, 46 | fnprb 7159 |
. . . . 5
⊢ (𝐹 Fn {𝐴, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) |
73 | | tpidm12 4717 |
. . . . . . 7
⊢ {𝐴, 𝐴, 𝐶} = {𝐴, 𝐶} |
74 | 73 | eqcomi 2746 |
. . . . . 6
⊢ {𝐴, 𝐶} = {𝐴, 𝐴, 𝐶} |
75 | 74 | fneq2i 6601 |
. . . . 5
⊢ (𝐹 Fn {𝐴, 𝐶} ↔ 𝐹 Fn {𝐴, 𝐴, 𝐶}) |
76 | | tpidm12 4717 |
. . . . . . 7
⊢
{⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} |
77 | 76 | eqcomi 2746 |
. . . . . 6
⊢
{⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} |
78 | 77 | eqeq2i 2750 |
. . . . 5
⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) |
79 | 72, 75, 78 | 3bitr3i 301 |
. . . 4
⊢ (𝐹 Fn {𝐴, 𝐴, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) |
80 | 79 | a1i 11 |
. . 3
⊢ (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐴, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})) |
81 | | tpeq2 4705 |
. . . 4
⊢ (𝐴 = 𝐵 → {𝐴, 𝐴, 𝐶} = {𝐴, 𝐵, 𝐶}) |
82 | 81 | fneq2d 6597 |
. . 3
⊢ (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐴, 𝐶} ↔ 𝐹 Fn {𝐴, 𝐵, 𝐶})) |
83 | | id 22 |
. . . . . 6
⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) |
84 | | fveq2 6843 |
. . . . . 6
⊢ (𝐴 = 𝐵 → (𝐹‘𝐴) = (𝐹‘𝐵)) |
85 | 83, 84 | opeq12d 4839 |
. . . . 5
⊢ (𝐴 = 𝐵 → ⟨𝐴, (𝐹‘𝐴)⟩ = ⟨𝐵, (𝐹‘𝐵)⟩) |
86 | 85 | tpeq2d 4708 |
. . . 4
⊢ (𝐴 = 𝐵 → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) |
87 | 86 | eqeq2d 2748 |
. . 3
⊢ (𝐴 = 𝐵 → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐶, (𝐹‘𝐶)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})) |
88 | 80, 82, 87 | 3bitr3d 309 |
. 2
⊢ (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})) |
89 | | tpidm13 4718 |
. . . . . . 7
⊢ {𝐴, 𝐵, 𝐴} = {𝐴, 𝐵} |
90 | 89 | eqcomi 2746 |
. . . . . 6
⊢ {𝐴, 𝐵} = {𝐴, 𝐵, 𝐴} |
91 | 90 | fneq2i 6601 |
. . . . 5
⊢ (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 Fn {𝐴, 𝐵, 𝐴}) |
92 | | tpidm13 4718 |
. . . . . . 7
⊢
{⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐴, (𝐹‘𝐴)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} |
93 | 92 | eqcomi 2746 |
. . . . . 6
⊢
{⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐴, (𝐹‘𝐴)⟩} |
94 | 93 | eqeq2i 2750 |
. . . . 5
⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐴, (𝐹‘𝐴)⟩}) |
95 | 3, 91, 94 | 3bitr3i 301 |
. . . 4
⊢ (𝐹 Fn {𝐴, 𝐵, 𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐴, (𝐹‘𝐴)⟩}) |
96 | 95 | a1i 11 |
. . 3
⊢ (𝐴 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐴, (𝐹‘𝐴)⟩})) |
97 | | tpeq3 4706 |
. . . 4
⊢ (𝐴 = 𝐶 → {𝐴, 𝐵, 𝐴} = {𝐴, 𝐵, 𝐶}) |
98 | 97 | fneq2d 6597 |
. . 3
⊢ (𝐴 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐴} ↔ 𝐹 Fn {𝐴, 𝐵, 𝐶})) |
99 | | id 22 |
. . . . . 6
⊢ (𝐴 = 𝐶 → 𝐴 = 𝐶) |
100 | | fveq2 6843 |
. . . . . 6
⊢ (𝐴 = 𝐶 → (𝐹‘𝐴) = (𝐹‘𝐶)) |
101 | 99, 100 | opeq12d 4839 |
. . . . 5
⊢ (𝐴 = 𝐶 → ⟨𝐴, (𝐹‘𝐴)⟩ = ⟨𝐶, (𝐹‘𝐶)⟩) |
102 | 101 | tpeq3d 4709 |
. . . 4
⊢ (𝐴 = 𝐶 → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐴, (𝐹‘𝐴)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) |
103 | 102 | eqeq2d 2748 |
. . 3
⊢ (𝐴 = 𝐶 → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐴, (𝐹‘𝐴)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})) |
104 | 96, 98, 103 | 3bitr3d 309 |
. 2
⊢ (𝐴 = 𝐶 → (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩})) |
105 | 71, 88, 104 | pm2.61iine 3036 |
1
⊢ (𝐹 Fn {𝐴, 𝐵, 𝐶} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩, ⟨𝐶, (𝐹‘𝐶)⟩}) |