Step | Hyp | Ref
| Expression |
1 | | fnse.3 |
. . . . . . 7
⊢ (𝜑 → 𝑅 Se 𝐵) |
2 | | fnse.2 |
. . . . . . . 8
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
3 | 2 | ffvelrnda 6904 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) ∈ 𝐵) |
4 | | seex 5513 |
. . . . . . 7
⊢ ((𝑅 Se 𝐵 ∧ (𝐹‘𝑧) ∈ 𝐵) → {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∈ V) |
5 | 1, 3, 4 | syl2an2r 685 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∈ V) |
6 | | snex 5324 |
. . . . . 6
⊢ {(𝐹‘𝑧)} ∈ V |
7 | | unexg 7534 |
. . . . . 6
⊢ (({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∈ V ∧ {(𝐹‘𝑧)} ∈ V) → ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V) |
8 | 5, 6, 7 | sylancl 589 |
. . . . 5
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V) |
9 | | imaeq2 5925 |
. . . . . . . . 9
⊢ (𝑤 = ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) → (◡𝐹 “ 𝑤) = (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))) |
10 | 9 | eleq1d 2822 |
. . . . . . . 8
⊢ (𝑤 = ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) → ((◡𝐹 “ 𝑤) ∈ V ↔ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V)) |
11 | 10 | imbi2d 344 |
. . . . . . 7
⊢ (𝑤 = ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) → ((𝜑 → (◡𝐹 “ 𝑤) ∈ V) ↔ (𝜑 → (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V))) |
12 | | fnse.4 |
. . . . . . 7
⊢ (𝜑 → (◡𝐹 “ 𝑤) ∈ V) |
13 | 11, 12 | vtoclg 3481 |
. . . . . 6
⊢ (({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V → (𝜑 → (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V)) |
14 | 13 | impcom 411 |
. . . . 5
⊢ ((𝜑 ∧ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ∈ V) → (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V) |
15 | 8, 14 | syldan 594 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ∈ V) |
16 | | inss2 4144 |
. . . . . 6
⊢ (𝐴 ∩ (◡𝑇 “ {𝑧})) ⊆ (◡𝑇 “ {𝑧}) |
17 | | vex 3412 |
. . . . . . . . . 10
⊢ 𝑤 ∈ V |
18 | 17 | eliniseg 5962 |
. . . . . . . . 9
⊢ (𝑧 ∈ V → (𝑤 ∈ (◡𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧)) |
19 | 18 | elv 3414 |
. . . . . . . 8
⊢ (𝑤 ∈ (◡𝑇 “ {𝑧}) ↔ 𝑤𝑇𝑧) |
20 | | fveq2 6717 |
. . . . . . . . . . . 12
⊢ (𝑥 = 𝑤 → (𝐹‘𝑥) = (𝐹‘𝑤)) |
21 | | fveq2 6717 |
. . . . . . . . . . . 12
⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧)) |
22 | 20, 21 | breqan12d 5069 |
. . . . . . . . . . 11
⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑧))) |
23 | 20, 21 | eqeqan12d 2751 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → ((𝐹‘𝑥) = (𝐹‘𝑦) ↔ (𝐹‘𝑤) = (𝐹‘𝑧))) |
24 | | breq12 5058 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (𝑥𝑆𝑦 ↔ 𝑤𝑆𝑧)) |
25 | 23, 24 | anbi12d 634 |
. . . . . . . . . . 11
⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦) ↔ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧))) |
26 | 22, 25 | orbi12d 919 |
. . . . . . . . . 10
⊢ ((𝑥 = 𝑤 ∧ 𝑦 = 𝑧) → (((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)) ↔ ((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧)))) |
27 | | fnse.1 |
. . . . . . . . . 10
⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ ((𝐹‘𝑥)𝑅(𝐹‘𝑦) ∨ ((𝐹‘𝑥) = (𝐹‘𝑦) ∧ 𝑥𝑆𝑦)))} |
28 | 26, 27 | brab2a 5641 |
. . . . . . . . 9
⊢ (𝑤𝑇𝑧 ↔ ((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧)))) |
29 | 2 | ffvelrnda 6904 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘𝑤) ∈ 𝐵) |
30 | 29 | adantrr 717 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝐹‘𝑤) ∈ 𝐵) |
31 | | breq1 5056 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑢 = (𝐹‘𝑤) → (𝑢𝑅(𝐹‘𝑧) ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑧))) |
32 | 31 | elrab3 3603 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹‘𝑤) ∈ 𝐵 → ((𝐹‘𝑤) ∈ {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑧))) |
33 | 30, 32 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑤) ∈ {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ↔ (𝐹‘𝑤)𝑅(𝐹‘𝑧))) |
34 | 33 | biimprd 251 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝐹‘𝑤)𝑅(𝐹‘𝑧) → (𝐹‘𝑤) ∈ {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)})) |
35 | | simpl 486 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧) → (𝐹‘𝑤) = (𝐹‘𝑧)) |
36 | | fvex 6730 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐹‘𝑤) ∈ V |
37 | 36 | elsn 4556 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐹‘𝑤) ∈ {(𝐹‘𝑧)} ↔ (𝐹‘𝑤) = (𝐹‘𝑧)) |
38 | 35, 37 | sylibr 237 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧) → (𝐹‘𝑤) ∈ {(𝐹‘𝑧)}) |
39 | 38 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧) → (𝐹‘𝑤) ∈ {(𝐹‘𝑧)})) |
40 | 34, 39 | orim12d 965 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧)) → ((𝐹‘𝑤) ∈ {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∨ (𝐹‘𝑤) ∈ {(𝐹‘𝑧)}))) |
41 | | elun 4063 |
. . . . . . . . . . . . 13
⊢ ((𝐹‘𝑤) ∈ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}) ↔ ((𝐹‘𝑤) ∈ {𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∨ (𝐹‘𝑤) ∈ {(𝐹‘𝑧)})) |
42 | 40, 41 | syl6ibr 255 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧)) → (𝐹‘𝑤) ∈ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))) |
43 | | simprl 771 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝑤 ∈ 𝐴) |
44 | 42, 43 | jctild 529 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧)) → (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) ∈ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))) |
45 | 2 | ffnd 6546 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐹 Fn 𝐴) |
46 | 45 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → 𝐹 Fn 𝐴) |
47 | | elpreima 6878 |
. . . . . . . . . . . 12
⊢ (𝐹 Fn 𝐴 → (𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) ∈ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))) |
48 | 46, 47 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})) ↔ (𝑤 ∈ 𝐴 ∧ (𝐹‘𝑤) ∈ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))) |
49 | 44, 48 | sylibrd 262 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧)) → 𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))) |
50 | 49 | expimpd 457 |
. . . . . . . . 9
⊢ (𝜑 → (((𝑤 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) ∧ ((𝐹‘𝑤)𝑅(𝐹‘𝑧) ∨ ((𝐹‘𝑤) = (𝐹‘𝑧) ∧ 𝑤𝑆𝑧))) → 𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))) |
51 | 28, 50 | syl5bi 245 |
. . . . . . . 8
⊢ (𝜑 → (𝑤𝑇𝑧 → 𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))) |
52 | 19, 51 | syl5bi 245 |
. . . . . . 7
⊢ (𝜑 → (𝑤 ∈ (◡𝑇 “ {𝑧}) → 𝑤 ∈ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)})))) |
53 | 52 | ssrdv 3907 |
. . . . . 6
⊢ (𝜑 → (◡𝑇 “ {𝑧}) ⊆ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))) |
54 | 16, 53 | sstrid 3912 |
. . . . 5
⊢ (𝜑 → (𝐴 ∩ (◡𝑇 “ {𝑧})) ⊆ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))) |
55 | 54 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐴 ∩ (◡𝑇 “ {𝑧})) ⊆ (◡𝐹 “ ({𝑢 ∈ 𝐵 ∣ 𝑢𝑅(𝐹‘𝑧)} ∪ {(𝐹‘𝑧)}))) |
56 | 15, 55 | ssexd 5217 |
. . 3
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐴) → (𝐴 ∩ (◡𝑇 “ {𝑧})) ∈ V) |
57 | 56 | ralrimiva 3105 |
. 2
⊢ (𝜑 → ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑇 “ {𝑧})) ∈ V) |
58 | | dfse2 5968 |
. 2
⊢ (𝑇 Se 𝐴 ↔ ∀𝑧 ∈ 𝐴 (𝐴 ∩ (◡𝑇 “ {𝑧})) ∈ V) |
59 | 57, 58 | sylibr 237 |
1
⊢ (𝜑 → 𝑇 Se 𝐴) |