Step | Hyp | Ref
| Expression |
1 | | fnprb.a |
. . . . . 6
⊢ 𝐴 ∈ V |
2 | 1 | fnsnb 7113 |
. . . . 5
⊢ (𝐹 Fn {𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩}) |
3 | | dfsn2 4600 |
. . . . . 6
⊢ {𝐴} = {𝐴, 𝐴} |
4 | 3 | fneq2i 6601 |
. . . . 5
⊢ (𝐹 Fn {𝐴} ↔ 𝐹 Fn {𝐴, 𝐴}) |
5 | | dfsn2 4600 |
. . . . . 6
⊢
{⟨𝐴, (𝐹‘𝐴)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩} |
6 | 5 | eqeq2i 2750 |
. . . . 5
⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩}) |
7 | 2, 4, 6 | 3bitr3i 301 |
. . . 4
⊢ (𝐹 Fn {𝐴, 𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩}) |
8 | 7 | a1i 11 |
. . 3
⊢ (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐴} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩})) |
9 | | preq2 4696 |
. . . 4
⊢ (𝐴 = 𝐵 → {𝐴, 𝐴} = {𝐴, 𝐵}) |
10 | 9 | fneq2d 6597 |
. . 3
⊢ (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐴} ↔ 𝐹 Fn {𝐴, 𝐵})) |
11 | | id 22 |
. . . . . 6
⊢ (𝐴 = 𝐵 → 𝐴 = 𝐵) |
12 | | fveq2 6843 |
. . . . . 6
⊢ (𝐴 = 𝐵 → (𝐹‘𝐴) = (𝐹‘𝐵)) |
13 | 11, 12 | opeq12d 4839 |
. . . . 5
⊢ (𝐴 = 𝐵 → ⟨𝐴, (𝐹‘𝐴)⟩ = ⟨𝐵, (𝐹‘𝐵)⟩) |
14 | 13 | preq2d 4702 |
. . . 4
⊢ (𝐴 = 𝐵 → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩} = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}) |
15 | 14 | eqeq2d 2748 |
. . 3
⊢ (𝐴 = 𝐵 → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐴, (𝐹‘𝐴)⟩} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) |
16 | 8, 10, 15 | 3bitr3d 309 |
. 2
⊢ (𝐴 = 𝐵 → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) |
17 | | fndm 6606 |
. . . . . 6
⊢ (𝐹 Fn {𝐴, 𝐵} → dom 𝐹 = {𝐴, 𝐵}) |
18 | | fvex 6856 |
. . . . . . 7
⊢ (𝐹‘𝐴) ∈ V |
19 | | fvex 6856 |
. . . . . . 7
⊢ (𝐹‘𝐵) ∈ V |
20 | 18, 19 | dmprop 6170 |
. . . . . 6
⊢ dom
{⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} = {𝐴, 𝐵} |
21 | 17, 20 | eqtr4di 2795 |
. . . . 5
⊢ (𝐹 Fn {𝐴, 𝐵} → dom 𝐹 = dom {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}) |
22 | 21 | adantl 483 |
. . . 4
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 Fn {𝐴, 𝐵}) → dom 𝐹 = dom {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}) |
23 | 17 | adantl 483 |
. . . . . . 7
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 Fn {𝐴, 𝐵}) → dom 𝐹 = {𝐴, 𝐵}) |
24 | 23 | eleq2d 2824 |
. . . . . 6
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 Fn {𝐴, 𝐵}) → (𝑥 ∈ dom 𝐹 ↔ 𝑥 ∈ {𝐴, 𝐵})) |
25 | | vex 3450 |
. . . . . . . 8
⊢ 𝑥 ∈ V |
26 | 25 | elpr 4610 |
. . . . . . 7
⊢ (𝑥 ∈ {𝐴, 𝐵} ↔ (𝑥 = 𝐴 ∨ 𝑥 = 𝐵)) |
27 | 1, 18 | fvpr1 7140 |
. . . . . . . . . . 11
⊢ (𝐴 ≠ 𝐵 → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐴) = (𝐹‘𝐴)) |
28 | 27 | adantr 482 |
. . . . . . . . . 10
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 Fn {𝐴, 𝐵}) → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐴) = (𝐹‘𝐴)) |
29 | 28 | eqcomd 2743 |
. . . . . . . . 9
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 Fn {𝐴, 𝐵}) → (𝐹‘𝐴) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐴)) |
30 | | fveq2 6843 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) |
31 | | fveq2 6843 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐴 → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐴)) |
32 | 30, 31 | eqeq12d 2753 |
. . . . . . . . 9
⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥) ↔ (𝐹‘𝐴) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐴))) |
33 | 29, 32 | syl5ibrcom 247 |
. . . . . . . 8
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 Fn {𝐴, 𝐵}) → (𝑥 = 𝐴 → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥))) |
34 | | fnprb.b |
. . . . . . . . . . . 12
⊢ 𝐵 ∈ V |
35 | 34, 19 | fvpr2 7142 |
. . . . . . . . . . 11
⊢ (𝐴 ≠ 𝐵 → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐵) = (𝐹‘𝐵)) |
36 | 35 | adantr 482 |
. . . . . . . . . 10
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 Fn {𝐴, 𝐵}) → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐵) = (𝐹‘𝐵)) |
37 | 36 | eqcomd 2743 |
. . . . . . . . 9
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 Fn {𝐴, 𝐵}) → (𝐹‘𝐵) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐵)) |
38 | | fveq2 6843 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐵 → (𝐹‘𝑥) = (𝐹‘𝐵)) |
39 | | fveq2 6843 |
. . . . . . . . . 10
⊢ (𝑥 = 𝐵 → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐵)) |
40 | 38, 39 | eqeq12d 2753 |
. . . . . . . . 9
⊢ (𝑥 = 𝐵 → ((𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥) ↔ (𝐹‘𝐵) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝐵))) |
41 | 37, 40 | syl5ibrcom 247 |
. . . . . . . 8
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 Fn {𝐴, 𝐵}) → (𝑥 = 𝐵 → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥))) |
42 | 33, 41 | jaod 858 |
. . . . . . 7
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 Fn {𝐴, 𝐵}) → ((𝑥 = 𝐴 ∨ 𝑥 = 𝐵) → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥))) |
43 | 26, 42 | biimtrid 241 |
. . . . . 6
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 Fn {𝐴, 𝐵}) → (𝑥 ∈ {𝐴, 𝐵} → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥))) |
44 | 24, 43 | sylbid 239 |
. . . . 5
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 Fn {𝐴, 𝐵}) → (𝑥 ∈ dom 𝐹 → (𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥))) |
45 | 44 | ralrimiv 3143 |
. . . 4
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 Fn {𝐴, 𝐵}) → ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥)) |
46 | | fnfun 6603 |
. . . . 5
⊢ (𝐹 Fn {𝐴, 𝐵} → Fun 𝐹) |
47 | 1, 34, 18, 19 | funpr 6558 |
. . . . 5
⊢ (𝐴 ≠ 𝐵 → Fun {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}) |
48 | | eqfunfv 6988 |
. . . . 5
⊢ ((Fun
𝐹 ∧ Fun {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}) → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} ↔ (dom 𝐹 = dom {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥)))) |
49 | 46, 47, 48 | syl2anr 598 |
. . . 4
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 Fn {𝐴, 𝐵}) → (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} ↔ (dom 𝐹 = dom {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} ∧ ∀𝑥 ∈ dom 𝐹(𝐹‘𝑥) = ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}‘𝑥)))) |
50 | 22, 45, 49 | mpbir2and 712 |
. . 3
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 Fn {𝐴, 𝐵}) → 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}) |
51 | | df-fn 6500 |
. . . . 5
⊢
({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} Fn {𝐴, 𝐵} ↔ (Fun {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} ∧ dom {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} = {𝐴, 𝐵})) |
52 | 47, 20, 51 | sylanblrc 591 |
. . . 4
⊢ (𝐴 ≠ 𝐵 → {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} Fn {𝐴, 𝐵}) |
53 | | fneq1 6594 |
. . . . 5
⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} → (𝐹 Fn {𝐴, 𝐵} ↔ {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} Fn {𝐴, 𝐵})) |
54 | 53 | biimprd 248 |
. . . 4
⊢ (𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} → ({⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩} Fn {𝐴, 𝐵} → 𝐹 Fn {𝐴, 𝐵})) |
55 | 52, 54 | mpan9 508 |
. . 3
⊢ ((𝐴 ≠ 𝐵 ∧ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}) → 𝐹 Fn {𝐴, 𝐵}) |
56 | 50, 55 | impbida 800 |
. 2
⊢ (𝐴 ≠ 𝐵 → (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩})) |
57 | 16, 56 | pm2.61ine 3029 |
1
⊢ (𝐹 Fn {𝐴, 𝐵} ↔ 𝐹 = {⟨𝐴, (𝐹‘𝐴)⟩, ⟨𝐵, (𝐹‘𝐵)⟩}) |