Step | Hyp | Ref
| Expression |
1 | | fpwrelmap.3 |
. . 3
⊢ 𝑀 = (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) |
2 | | fpwrelmap.1 |
. . . . . 6
⊢ 𝐴 ∈ V |
3 | 2 | a1i 11 |
. . . . 5
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → 𝐴 ∈ V) |
4 | | simpr 477 |
. . . . . . . . 9
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑦 ∈ (𝑓‘𝑥)) |
5 | | elmapi 7879 |
. . . . . . . . . . 11
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → 𝑓:𝐴⟶𝒫 𝐵) |
6 | 5 | ffvelrnda 6359 |
. . . . . . . . . 10
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ 𝒫 𝐵) |
7 | 6 | adantr 481 |
. . . . . . . . 9
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → (𝑓‘𝑥) ∈ 𝒫 𝐵) |
8 | | elelpwi 4171 |
. . . . . . . . 9
⊢ ((𝑦 ∈ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ∈ 𝒫 𝐵) → 𝑦 ∈ 𝐵) |
9 | 4, 7, 8 | syl2anc 693 |
. . . . . . . 8
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑦 ∈ 𝐵) |
10 | 9 | ex 450 |
. . . . . . 7
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) → 𝑦 ∈ 𝐵)) |
11 | 10 | alrimiv 1855 |
. . . . . 6
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 ∈ (𝑓‘𝑥) → 𝑦 ∈ 𝐵)) |
12 | | abss 3671 |
. . . . . . 7
⊢ ({𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ (𝑓‘𝑥) → 𝑦 ∈ 𝐵)) |
13 | | fpwrelmap.2 |
. . . . . . . 8
⊢ 𝐵 ∈ V |
14 | 13 | ssex 4802 |
. . . . . . 7
⊢ ({𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} ⊆ 𝐵 → {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} ∈ V) |
15 | 12, 14 | sylbir 225 |
. . . . . 6
⊢
(∀𝑦(𝑦 ∈ (𝑓‘𝑥) → 𝑦 ∈ 𝐵) → {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} ∈ V) |
16 | 11, 15 | syl 17 |
. . . . 5
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} ∈ V) |
17 | 3, 16 | opabex3d 7145 |
. . . 4
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ∈ V) |
18 | 17 | adantl 482 |
. . 3
⊢
((⊤ ∧ 𝑓
∈ (𝒫 𝐵
↑𝑚 𝐴)) → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ∈ V) |
19 | 2 | mptex 6486 |
. . . 4
⊢ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) ∈ V |
20 | 19 | a1i 11 |
. . 3
⊢
((⊤ ∧ 𝑟
∈ 𝒫 (𝐴 ×
𝐵)) → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) ∈ V) |
21 | 10 | imdistanda 729 |
. . . . . . . . . 10
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
22 | 21 | ssopab2dv 5004 |
. . . . . . . . 9
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}) |
23 | 22 | adantr 481 |
. . . . . . . 8
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}) |
24 | | simpr 477 |
. . . . . . . 8
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) |
25 | | df-xp 5120 |
. . . . . . . . 9
⊢ (𝐴 × 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)} |
26 | 25 | a1i 11 |
. . . . . . . 8
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → (𝐴 × 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}) |
27 | 23, 24, 26 | 3sstr4d 3648 |
. . . . . . 7
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑟 ⊆ (𝐴 × 𝐵)) |
28 | | selpw 4165 |
. . . . . . 7
⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑟 ⊆ (𝐴 × 𝐵)) |
29 | 27, 28 | sylibr 224 |
. . . . . 6
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) |
30 | 5 | feqmptd 6249 |
. . . . . . . 8
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥))) |
31 | 30 | adantr 481 |
. . . . . . 7
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥))) |
32 | | nfv 1843 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) |
33 | | nfopab1 4719 |
. . . . . . . . . 10
⊢
Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} |
34 | 33 | nfeq2 2780 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} |
35 | 32, 34 | nfan 1828 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) |
36 | | df-rab 2921 |
. . . . . . . . . 10
⊢ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦)} |
37 | 36 | a1i 11 |
. . . . . . . . 9
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦)}) |
38 | | nfv 1843 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑦 𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) |
39 | | nfopab2 4720 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} |
40 | 39 | nfeq2 2780 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑦 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} |
41 | 38, 40 | nfan 1828 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦(𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) |
42 | | nfv 1843 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦 𝑥 ∈ 𝐴 |
43 | 41, 42 | nfan 1828 |
. . . . . . . . . 10
⊢
Ⅎ𝑦((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) |
44 | 9 | adantllr 755 |
. . . . . . . . . . . . 13
⊢ ((((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑦 ∈ 𝐵) |
45 | | df-br 4654 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥𝑟𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑟) |
46 | | eleq2 2690 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} → (〈𝑥, 𝑦〉 ∈ 𝑟 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})) |
47 | | opabid 4982 |
. . . . . . . . . . . . . . . . . 18
⊢
(〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))) |
48 | 46, 47 | syl6bb 276 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} → (〈𝑥, 𝑦〉 ∈ 𝑟 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
49 | 45, 48 | syl5bb 272 |
. . . . . . . . . . . . . . . 16
⊢ (𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} → (𝑥𝑟𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
50 | 49 | ad2antlr 763 |
. . . . . . . . . . . . . . 15
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑥𝑟𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
51 | | elfvdm 6220 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (𝑓‘𝑥) → 𝑥 ∈ dom 𝑓) |
52 | 51 | adantl 482 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑥 ∈ dom 𝑓) |
53 | | fdm 6051 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑓:𝐴⟶𝒫 𝐵 → dom 𝑓 = 𝐴) |
54 | 5, 53 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → dom 𝑓 = 𝐴) |
55 | 54 | adantr 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → dom 𝑓 = 𝐴) |
56 | 52, 55 | eleqtrd 2703 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑥 ∈ 𝐴) |
57 | 56 | ex 450 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → (𝑦 ∈ (𝑓‘𝑥) → 𝑥 ∈ 𝐴)) |
58 | 57 | pm4.71rd 667 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → (𝑦 ∈ (𝑓‘𝑥) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
59 | 58 | ad2antrr 762 |
. . . . . . . . . . . . . . 15
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
60 | 50, 59 | bitr4d 271 |
. . . . . . . . . . . . . 14
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑥𝑟𝑦 ↔ 𝑦 ∈ (𝑓‘𝑥))) |
61 | 60 | biimpar 502 |
. . . . . . . . . . . . 13
⊢ ((((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑥𝑟𝑦) |
62 | 44, 61 | jca 554 |
. . . . . . . . . . . 12
⊢ ((((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦)) |
63 | 62 | ex 450 |
. . . . . . . . . . 11
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) → (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦))) |
64 | 60 | biimpd 219 |
. . . . . . . . . . . 12
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑥𝑟𝑦 → 𝑦 ∈ (𝑓‘𝑥))) |
65 | 64 | adantld 483 |
. . . . . . . . . . 11
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦) → 𝑦 ∈ (𝑓‘𝑥))) |
66 | 63, 65 | impbid 202 |
. . . . . . . . . 10
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦))) |
67 | 43, 66 | abbid 2740 |
. . . . . . . . 9
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦)}) |
68 | | abid2 2745 |
. . . . . . . . . 10
⊢ {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} = (𝑓‘𝑥) |
69 | 68 | a1i 11 |
. . . . . . . . 9
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} = (𝑓‘𝑥)) |
70 | 37, 67, 69 | 3eqtr2rd 2663 |
. . . . . . . 8
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
71 | 35, 70 | mpteq2da 4743 |
. . . . . . 7
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
72 | 31, 71 | eqtrd 2656 |
. . . . . 6
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
73 | 29, 72 | jca 554 |
. . . . 5
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) |
74 | | ssrab2 3687 |
. . . . . . . . . . . 12
⊢ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ⊆ 𝐵 |
75 | 13 | elpw2 4828 |
. . . . . . . . . . . 12
⊢ ({𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ 𝒫 𝐵 ↔ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ⊆ 𝐵) |
76 | 74, 75 | mpbir 221 |
. . . . . . . . . . 11
⊢ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ 𝒫 𝐵 |
77 | 76 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐴) → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ 𝒫 𝐵) |
78 | | eqid 2622 |
. . . . . . . . . 10
⊢ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
79 | 77, 78 | fmptd 6385 |
. . . . . . . . 9
⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵) |
80 | 79 | adantr 481 |
. . . . . . . 8
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵) |
81 | | simpr 477 |
. . . . . . . . 9
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
82 | 81 | feq1d 6030 |
. . . . . . . 8
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑓:𝐴⟶𝒫 𝐵 ↔ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)) |
83 | 80, 82 | mpbird 247 |
. . . . . . 7
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑓:𝐴⟶𝒫 𝐵) |
84 | 13 | pwex 4848 |
. . . . . . . 8
⊢ 𝒫
𝐵 ∈ V |
85 | 84, 2 | elmap 7886 |
. . . . . . 7
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵) |
86 | 83, 85 | sylibr 224 |
. . . . . 6
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) |
87 | | elpwi 4168 |
. . . . . . . . . 10
⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → 𝑟 ⊆ (𝐴 × 𝐵)) |
88 | 87 | adantr 481 |
. . . . . . . . 9
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵)) |
89 | | xpss 5226 |
. . . . . . . . 9
⊢ (𝐴 × 𝐵) ⊆ (V × V) |
90 | 88, 89 | syl6ss 3615 |
. . . . . . . 8
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ⊆ (V × V)) |
91 | | df-rel 5121 |
. . . . . . . 8
⊢ (Rel
𝑟 ↔ 𝑟 ⊆ (V × V)) |
92 | 90, 91 | sylibr 224 |
. . . . . . 7
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → Rel 𝑟) |
93 | | relopab 5247 |
. . . . . . . 8
⊢ Rel
{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} |
94 | 93 | a1i 11 |
. . . . . . 7
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → Rel {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) |
95 | | id 22 |
. . . . . . 7
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) |
96 | | nfv 1843 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑟 ∈ 𝒫 (𝐴 × 𝐵) |
97 | | nfmpt1 4747 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
98 | 97 | nfeq2 2780 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
99 | 96, 98 | nfan 1828 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
100 | | nfv 1843 |
. . . . . . . . 9
⊢
Ⅎ𝑦 𝑟 ∈ 𝒫 (𝐴 × 𝐵) |
101 | 42 | nfci 2754 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦𝐴 |
102 | | nfrab1 3122 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} |
103 | 101, 102 | nfmpt 4746 |
. . . . . . . . . 10
⊢
Ⅎ𝑦(𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
104 | 103 | nfeq2 2780 |
. . . . . . . . 9
⊢
Ⅎ𝑦 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
105 | 100, 104 | nfan 1828 |
. . . . . . . 8
⊢
Ⅎ𝑦(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
106 | | nfcv 2764 |
. . . . . . . 8
⊢
Ⅎ𝑥𝑟 |
107 | | nfcv 2764 |
. . . . . . . 8
⊢
Ⅎ𝑦𝑟 |
108 | | brelg 29421 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑟 ⊆ (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
109 | 87, 108 | sylan 488 |
. . . . . . . . . . . . . . 15
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
110 | 109 | adantlr 751 |
. . . . . . . . . . . . . 14
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
111 | 110 | simpld 475 |
. . . . . . . . . . . . 13
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥 ∈ 𝐴) |
112 | 110 | simprd 479 |
. . . . . . . . . . . . . 14
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦 ∈ 𝐵) |
113 | | simpr 477 |
. . . . . . . . . . . . . 14
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥𝑟𝑦) |
114 | 81 | fveq1d 6193 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑓‘𝑥) = ((𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})‘𝑥)) |
115 | 13 | rabex 4813 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ V |
116 | 78 | fvmpt2 6291 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ 𝐴 ∧ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ V) → ((𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})‘𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
117 | 115, 116 | mpan2 707 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})‘𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
118 | 114, 117 | sylan9eq 2676 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
119 | 118 | eleq2d 2687 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) ↔ 𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
120 | | rabid 3116 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ↔ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦)) |
121 | 119, 120 | syl6bb 276 |
. . . . . . . . . . . . . . 15
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦))) |
122 | 111, 121 | syldan 487 |
. . . . . . . . . . . . . 14
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑦 ∈ (𝑓‘𝑥) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦))) |
123 | 112, 113,
122 | mpbir2and 957 |
. . . . . . . . . . . . 13
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦 ∈ (𝑓‘𝑥)) |
124 | 111, 123 | jca 554 |
. . . . . . . . . . . 12
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))) |
125 | 124 | ex 450 |
. . . . . . . . . . 11
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑥𝑟𝑦 → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
126 | 121 | simplbda 654 |
. . . . . . . . . . . 12
⊢ ((((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑥𝑟𝑦) |
127 | 126 | expl 648 |
. . . . . . . . . . 11
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑥𝑟𝑦)) |
128 | 125, 127 | impbid 202 |
. . . . . . . . . 10
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑥𝑟𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
129 | 45, 128 | syl5bbr 274 |
. . . . . . . . 9
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (〈𝑥, 𝑦〉 ∈ 𝑟 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
130 | 129, 47 | syl6bbr 278 |
. . . . . . . 8
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (〈𝑥, 𝑦〉 ∈ 𝑟 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})) |
131 | 99, 105, 106, 107, 33, 39, 130 | eqrelrd2 29426 |
. . . . . . 7
⊢ (((Rel
𝑟 ∧ Rel {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) → 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) |
132 | 92, 94, 95, 131 | syl21anc 1325 |
. . . . . 6
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) |
133 | 86, 132 | jca 554 |
. . . . 5
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})) |
134 | 73, 133 | impbii 199 |
. . . 4
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) |
135 | 134 | a1i 11 |
. . 3
⊢ (⊤
→ ((𝑓 ∈
(𝒫 𝐵
↑𝑚 𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})))) |
136 | 1, 18, 20, 135 | f1od 6885 |
. 2
⊢ (⊤
→ 𝑀:(𝒫 𝐵 ↑𝑚
𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)) |
137 | 136 | trud 1493 |
1
⊢ 𝑀:(𝒫 𝐵 ↑𝑚 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵) |