Step | Hyp | Ref
| Expression |
1 | | fundmen.1 |
. . . 4
⊢ 𝐹 ∈ V |
2 | 1 | dmex 7848 |
. . 3
⊢ dom 𝐹 ∈ V |
3 | 2 | a1i 11 |
. 2
⊢ (Fun
𝐹 → dom 𝐹 ∈ V) |
4 | 1 | a1i 11 |
. 2
⊢ (Fun
𝐹 → 𝐹 ∈ V) |
5 | | funfvop 7000 |
. . 3
⊢ ((Fun
𝐹 ∧ 𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹) |
6 | 5 | ex 413 |
. 2
⊢ (Fun
𝐹 → (𝑥 ∈ dom 𝐹 → ⟨𝑥, (𝐹‘𝑥)⟩ ∈ 𝐹)) |
7 | | funrel 6518 |
. . 3
⊢ (Fun
𝐹 → Rel 𝐹) |
8 | | elreldm 5890 |
. . . 4
⊢ ((Rel
𝐹 ∧ 𝑦 ∈ 𝐹) → ∩ ∩ 𝑦
∈ dom 𝐹) |
9 | 8 | ex 413 |
. . 3
⊢ (Rel
𝐹 → (𝑦 ∈ 𝐹 → ∩ ∩ 𝑦
∈ dom 𝐹)) |
10 | 7, 9 | syl 17 |
. 2
⊢ (Fun
𝐹 → (𝑦 ∈ 𝐹 → ∩ ∩ 𝑦
∈ dom 𝐹)) |
11 | | df-rel 5640 |
. . . . . . . . 9
⊢ (Rel
𝐹 ↔ 𝐹 ⊆ (V × V)) |
12 | 7, 11 | sylib 217 |
. . . . . . . 8
⊢ (Fun
𝐹 → 𝐹 ⊆ (V × V)) |
13 | 12 | sselda 3944 |
. . . . . . 7
⊢ ((Fun
𝐹 ∧ 𝑦 ∈ 𝐹) → 𝑦 ∈ (V × V)) |
14 | | elvv 5706 |
. . . . . . 7
⊢ (𝑦 ∈ (V × V) ↔
∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩) |
15 | 13, 14 | sylib 217 |
. . . . . 6
⊢ ((Fun
𝐹 ∧ 𝑦 ∈ 𝐹) → ∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩) |
16 | | inteq 4910 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩
𝑦 = ∩ ⟨𝑧, 𝑤⟩) |
17 | 16 | inteqd 4912 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩
∩ 𝑦 = ∩ ∩ ⟨𝑧, 𝑤⟩) |
18 | | vex 3449 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑧 ∈ V |
19 | | vex 3449 |
. . . . . . . . . . . . . . . . 17
⊢ 𝑤 ∈ V |
20 | 18, 19 | op1stb 5428 |
. . . . . . . . . . . . . . . 16
⊢ ∩ ∩ ⟨𝑧, 𝑤⟩ = 𝑧 |
21 | 17, 20 | eqtrdi 2792 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → ∩
∩ 𝑦 = 𝑧) |
22 | | eqeq1 2740 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = ∩
∩ 𝑦 → (𝑥 = 𝑧 ↔ ∩ ∩ 𝑦 =
𝑧)) |
23 | 21, 22 | syl5ibr 245 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = ∩
∩ 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → 𝑥 = 𝑧)) |
24 | | opeq1 4830 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑧 → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩) |
25 | 23, 24 | syl6 35 |
. . . . . . . . . . . . 13
⊢ (𝑥 = ∩
∩ 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩)) |
26 | 25 | imp 407 |
. . . . . . . . . . . 12
⊢ ((𝑥 = ∩
∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → ⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩) |
27 | | eqeq2 2748 |
. . . . . . . . . . . . . 14
⊢
(⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → (𝑦 = ⟨𝑥, 𝑤⟩ ↔ 𝑦 = ⟨𝑧, 𝑤⟩)) |
28 | 27 | biimprcd 249 |
. . . . . . . . . . . . 13
⊢ (𝑦 = ⟨𝑧, 𝑤⟩ → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩)) |
29 | 28 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((𝑥 = ∩
∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → (⟨𝑥, 𝑤⟩ = ⟨𝑧, 𝑤⟩ → 𝑦 = ⟨𝑥, 𝑤⟩)) |
30 | 26, 29 | mpd 15 |
. . . . . . . . . . 11
⊢ ((𝑥 = ∩
∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → 𝑦 = ⟨𝑥, 𝑤⟩) |
31 | 30 | ancoms 459 |
. . . . . . . . . 10
⊢ ((𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦)
→ 𝑦 = ⟨𝑥, 𝑤⟩) |
32 | 31 | adantl 482 |
. . . . . . . . 9
⊢ (((Fun
𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦))
→ 𝑦 = ⟨𝑥, 𝑤⟩) |
33 | 30 | eleq1d 2822 |
. . . . . . . . . . . . . . 15
⊢ ((𝑥 = ∩
∩ 𝑦 ∧ 𝑦 = ⟨𝑧, 𝑤⟩) → (𝑦 ∈ 𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹)) |
34 | 33 | adantl 482 |
. . . . . . . . . . . . . 14
⊢ ((Fun
𝐹 ∧ (𝑥 = ∩ ∩ 𝑦
∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦 ∈ 𝐹 ↔ ⟨𝑥, 𝑤⟩ ∈ 𝐹)) |
35 | | funopfv 6894 |
. . . . . . . . . . . . . . 15
⊢ (Fun
𝐹 → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹‘𝑥) = 𝑤)) |
36 | 35 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ ((Fun
𝐹 ∧ (𝑥 = ∩ ∩ 𝑦
∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (⟨𝑥, 𝑤⟩ ∈ 𝐹 → (𝐹‘𝑥) = 𝑤)) |
37 | 34, 36 | sylbid 239 |
. . . . . . . . . . . . 13
⊢ ((Fun
𝐹 ∧ (𝑥 = ∩ ∩ 𝑦
∧ 𝑦 = ⟨𝑧, 𝑤⟩)) → (𝑦 ∈ 𝐹 → (𝐹‘𝑥) = 𝑤)) |
38 | 37 | exp32 421 |
. . . . . . . . . . . 12
⊢ (Fun
𝐹 → (𝑥 = ∩
∩ 𝑦 → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑦 ∈ 𝐹 → (𝐹‘𝑥) = 𝑤)))) |
39 | 38 | com24 95 |
. . . . . . . . . . 11
⊢ (Fun
𝐹 → (𝑦 ∈ 𝐹 → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = ∩ ∩ 𝑦
→ (𝐹‘𝑥) = 𝑤)))) |
40 | 39 | imp43 428 |
. . . . . . . . . 10
⊢ (((Fun
𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦))
→ (𝐹‘𝑥) = 𝑤) |
41 | 40 | opeq2d 4837 |
. . . . . . . . 9
⊢ (((Fun
𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦))
→ ⟨𝑥, (𝐹‘𝑥)⟩ = ⟨𝑥, 𝑤⟩) |
42 | 32, 41 | eqtr4d 2779 |
. . . . . . . 8
⊢ (((Fun
𝐹 ∧ 𝑦 ∈ 𝐹) ∧ (𝑦 = ⟨𝑧, 𝑤⟩ ∧ 𝑥 = ∩ ∩ 𝑦))
→ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩) |
43 | 42 | exp32 421 |
. . . . . . 7
⊢ ((Fun
𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = ∩ ∩ 𝑦
→ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩))) |
44 | 43 | exlimdvv 1937 |
. . . . . 6
⊢ ((Fun
𝐹 ∧ 𝑦 ∈ 𝐹) → (∃𝑧∃𝑤 𝑦 = ⟨𝑧, 𝑤⟩ → (𝑥 = ∩ ∩ 𝑦
→ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩))) |
45 | 15, 44 | mpd 15 |
. . . . 5
⊢ ((Fun
𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 = ∩ ∩ 𝑦
→ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)) |
46 | 45 | adantrl 714 |
. . . 4
⊢ ((Fun
𝐹 ∧ (𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 = ∩ ∩ 𝑦
→ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)) |
47 | | inteq 4910 |
. . . . . 6
⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → ∩
𝑦 = ∩ ⟨𝑥, (𝐹‘𝑥)⟩) |
48 | 47 | inteqd 4912 |
. . . . 5
⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → ∩
∩ 𝑦 = ∩ ∩ ⟨𝑥, (𝐹‘𝑥)⟩) |
49 | | vex 3449 |
. . . . . 6
⊢ 𝑥 ∈ V |
50 | | fvex 6855 |
. . . . . 6
⊢ (𝐹‘𝑥) ∈ V |
51 | 49, 50 | op1stb 5428 |
. . . . 5
⊢ ∩ ∩ ⟨𝑥, (𝐹‘𝑥)⟩ = 𝑥 |
52 | 48, 51 | eqtr2di 2793 |
. . . 4
⊢ (𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩ → 𝑥 = ∩ ∩ 𝑦) |
53 | 46, 52 | impbid1 224 |
. . 3
⊢ ((Fun
𝐹 ∧ (𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹)) → (𝑥 = ∩ ∩ 𝑦
↔ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩)) |
54 | 53 | ex 413 |
. 2
⊢ (Fun
𝐹 → ((𝑥 ∈ dom 𝐹 ∧ 𝑦 ∈ 𝐹) → (𝑥 = ∩ ∩ 𝑦
↔ 𝑦 = ⟨𝑥, (𝐹‘𝑥)⟩))) |
55 | 3, 4, 6, 10, 54 | en3d 8929 |
1
⊢ (Fun
𝐹 → dom 𝐹 ≈ 𝐹) |