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 8226 |
. . . . . . . . . . 11
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → 𝑓:𝐴⟶𝒫 𝐵) |
6 | 5 | ffvelrnda 6674 |
. . . . . . . . . 10
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ 𝒫 𝐵) |
7 | 6 | adantr 473 |
. . . . . . . . 9
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → (𝑓‘𝑥) ∈ 𝒫 𝐵) |
8 | | elelpwi 4429 |
. . . . . . . . 9
⊢ ((𝑦 ∈ (𝑓‘𝑥) ∧ (𝑓‘𝑥) ∈ 𝒫 𝐵) → 𝑦 ∈ 𝐵) |
9 | 4, 7, 8 | syl2anc 576 |
. . . . . . . 8
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑦 ∈ 𝐵) |
10 | 9 | ex 405 |
. . . . . . 7
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) → 𝑦 ∈ 𝐵)) |
11 | 10 | alrimiv 1886 |
. . . . . 6
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑥 ∈ 𝐴) → ∀𝑦(𝑦 ∈ (𝑓‘𝑥) → 𝑦 ∈ 𝐵)) |
12 | | abss 3924 |
. . . . . . 7
⊢ ({𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} ⊆ 𝐵 ↔ ∀𝑦(𝑦 ∈ (𝑓‘𝑥) → 𝑦 ∈ 𝐵)) |
13 | | fpwrelmap.2 |
. . . . . . . 8
⊢ 𝐵 ∈ V |
14 | 13 | ssex 5077 |
. . . . . . 7
⊢ ({𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} ⊆ 𝐵 → {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} ∈ V) |
15 | 12, 14 | sylbir 227 |
. . . . . 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 7476 |
. . . 4
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ∈ V) |
18 | 17 | adantl 474 |
. . 3
⊢
((⊤ ∧ 𝑓
∈ (𝒫 𝐵
↑𝑚 𝐴)) → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ∈ V) |
19 | 2 | mptex 6810 |
. . . 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 564 |
. . . . . . . . . 10
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
22 | 21 | ssopab2dv 5286 |
. . . . . . . . 9
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ⊆ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}) |
23 | 22 | adantr 473 |
. . . . . . . 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 5409 |
. . . . . . . . 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 3898 |
. . . . . . 7
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑟 ⊆ (𝐴 × 𝐵)) |
28 | | selpw 4423 |
. . . . . . 7
⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↔ 𝑟 ⊆ (𝐴 × 𝐵)) |
29 | 27, 28 | sylibr 226 |
. . . . . 6
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) |
30 | 5 | feqmptd 6560 |
. . . . . . . 8
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥))) |
31 | 30 | adantr 473 |
. . . . . . 7
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥))) |
32 | | nfv 1873 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) |
33 | | nfopab1 4994 |
. . . . . . . . . 10
⊢
Ⅎ𝑥{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} |
34 | 33 | nfeq2 2941 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} |
35 | 32, 34 | nfan 1862 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) |
36 | | df-rab 3091 |
. . . . . . . . . 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 1873 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑦 𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) |
39 | | nfopab2 4995 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑦{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} |
40 | 39 | nfeq2 2941 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑦 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} |
41 | 38, 40 | nfan 1862 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦(𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) |
42 | | nfv 1873 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦 𝑥 ∈ 𝐴 |
43 | 41, 42 | nfan 1862 |
. . . . . . . . . 10
⊢
Ⅎ𝑦((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) |
44 | 9 | adantllr 706 |
. . . . . . . . . . . . 13
⊢ ((((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑦 ∈ 𝐵) |
45 | | df-br 4926 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥𝑟𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝑟) |
46 | | eleq2 2848 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} → (〈𝑥, 𝑦〉 ∈ 𝑟 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})) |
47 | | opabid 5264 |
. . . . . . . . . . . . . . . . . 18
⊢
(〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))) |
48 | 46, 47 | syl6bb 279 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} → (〈𝑥, 𝑦〉 ∈ 𝑟 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
49 | 45, 48 | syl5bb 275 |
. . . . . . . . . . . . . . . 16
⊢ (𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} → (𝑥𝑟𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
50 | 49 | ad2antlr 714 |
. . . . . . . . . . . . . . 15
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑥𝑟𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
51 | | elfvdm 6528 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑦 ∈ (𝑓‘𝑥) → 𝑥 ∈ dom 𝑓) |
52 | 51 | adantl 474 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑥 ∈ dom 𝑓) |
53 | 5 | fdmd 6350 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → dom 𝑓 = 𝐴) |
54 | 53 | adantr 473 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → dom 𝑓 = 𝐴) |
55 | 52, 54 | eleqtrd 2862 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑥 ∈ 𝐴) |
56 | 55 | ex 405 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → (𝑦 ∈ (𝑓‘𝑥) → 𝑥 ∈ 𝐴)) |
57 | 56 | pm4.71rd 555 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) → (𝑦 ∈ (𝑓‘𝑥) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
58 | 57 | ad2antrr 713 |
. . . . . . . . . . . . . . 15
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
59 | 50, 58 | bitr4d 274 |
. . . . . . . . . . . . . 14
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑥𝑟𝑦 ↔ 𝑦 ∈ (𝑓‘𝑥))) |
60 | 59 | biimpar 470 |
. . . . . . . . . . . . 13
⊢ ((((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑥𝑟𝑦) |
61 | 44, 60 | jca 504 |
. . . . . . . . . . . 12
⊢ ((((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦)) |
62 | 61 | ex 405 |
. . . . . . . . . . 11
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) → (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦))) |
63 | 59 | biimpd 221 |
. . . . . . . . . . . 12
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑥𝑟𝑦 → 𝑦 ∈ (𝑓‘𝑥))) |
64 | 63 | adantld 483 |
. . . . . . . . . . 11
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → ((𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦) → 𝑦 ∈ (𝑓‘𝑥))) |
65 | 62, 64 | impbid 204 |
. . . . . . . . . 10
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦))) |
66 | 43, 65 | abbid 2839 |
. . . . . . . . 9
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} = {𝑦 ∣ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦)}) |
67 | | abid2 2903 |
. . . . . . . . . 10
⊢ {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} = (𝑓‘𝑥) |
68 | 67 | a1i 11 |
. . . . . . . . 9
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → {𝑦 ∣ 𝑦 ∈ (𝑓‘𝑥)} = (𝑓‘𝑥)) |
69 | 37, 66, 68 | 3eqtr2rd 2815 |
. . . . . . . 8
⊢ (((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
70 | 35, 69 | mpteq2da 5017 |
. . . . . . 7
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
71 | 31, 70 | eqtrd 2808 |
. . . . . 6
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
72 | 29, 71 | jca 504 |
. . . . 5
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) |
73 | | ssrab2 3940 |
. . . . . . . . . . . 12
⊢ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ⊆ 𝐵 |
74 | 13 | elpw2 5100 |
. . . . . . . . . . . 12
⊢ ({𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ 𝒫 𝐵 ↔ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ⊆ 𝐵) |
75 | 73, 74 | mpbir 223 |
. . . . . . . . . . 11
⊢ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ 𝒫 𝐵 |
76 | 75 | a1i 11 |
. . . . . . . . . 10
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥 ∈ 𝐴) → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ 𝒫 𝐵) |
77 | 76 | fmpttd 6700 |
. . . . . . . . 9
⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵) |
78 | 77 | adantr 473 |
. . . . . . . 8
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵) |
79 | | simpr 477 |
. . . . . . . . 9
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
80 | 79 | feq1d 6326 |
. . . . . . . 8
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑓:𝐴⟶𝒫 𝐵 ↔ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)) |
81 | 78, 80 | mpbird 249 |
. . . . . . 7
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑓:𝐴⟶𝒫 𝐵) |
82 | 13 | pwex 5130 |
. . . . . . . 8
⊢ 𝒫
𝐵 ∈ V |
83 | 82, 2 | elmap 8233 |
. . . . . . 7
⊢ (𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵) |
84 | 81, 83 | sylibr 226 |
. . . . . 6
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴)) |
85 | | elpwi 4426 |
. . . . . . . . . 10
⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) → 𝑟 ⊆ (𝐴 × 𝐵)) |
86 | 85 | adantr 473 |
. . . . . . . . 9
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵)) |
87 | | xpss 5419 |
. . . . . . . . 9
⊢ (𝐴 × 𝐵) ⊆ (V × V) |
88 | 86, 87 | syl6ss 3864 |
. . . . . . . 8
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ⊆ (V × V)) |
89 | | df-rel 5410 |
. . . . . . . 8
⊢ (Rel
𝑟 ↔ 𝑟 ⊆ (V × V)) |
90 | 88, 89 | sylibr 226 |
. . . . . . 7
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → Rel 𝑟) |
91 | | relopab 5542 |
. . . . . . . 8
⊢ Rel
{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} |
92 | 91 | a1i 11 |
. . . . . . 7
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → Rel {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) |
93 | | id 22 |
. . . . . . 7
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) |
94 | | nfv 1873 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑟 ∈ 𝒫 (𝐴 × 𝐵) |
95 | | nfmpt1 5021 |
. . . . . . . . . 10
⊢
Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
96 | 95 | nfeq2 2941 |
. . . . . . . . 9
⊢
Ⅎ𝑥 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
97 | 94, 96 | nfan 1862 |
. . . . . . . 8
⊢
Ⅎ𝑥(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
98 | | nfv 1873 |
. . . . . . . . 9
⊢
Ⅎ𝑦 𝑟 ∈ 𝒫 (𝐴 × 𝐵) |
99 | 42 | nfci 2913 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦𝐴 |
100 | | nfrab1 3318 |
. . . . . . . . . . 11
⊢
Ⅎ𝑦{𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} |
101 | 99, 100 | nfmpt 5020 |
. . . . . . . . . 10
⊢
Ⅎ𝑦(𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
102 | 101 | nfeq2 2941 |
. . . . . . . . 9
⊢
Ⅎ𝑦 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
103 | 98, 102 | nfan 1862 |
. . . . . . . 8
⊢
Ⅎ𝑦(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
104 | | nfcv 2926 |
. . . . . . . 8
⊢
Ⅎ𝑥𝑟 |
105 | | nfcv 2926 |
. . . . . . . 8
⊢
Ⅎ𝑦𝑟 |
106 | | brelg 30138 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑟 ⊆ (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
107 | 85, 106 | sylan 572 |
. . . . . . . . . . . . . . 15
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑥𝑟𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
108 | 107 | adantlr 702 |
. . . . . . . . . . . . . 14
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) |
109 | 108 | simpld 487 |
. . . . . . . . . . . . 13
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥 ∈ 𝐴) |
110 | 108 | simprd 488 |
. . . . . . . . . . . . . 14
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦 ∈ 𝐵) |
111 | | simpr 477 |
. . . . . . . . . . . . . 14
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑥𝑟𝑦) |
112 | 79 | fveq1d 6498 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑓‘𝑥) = ((𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})‘𝑥)) |
113 | 13 | rabex 5087 |
. . . . . . . . . . . . . . . . . . 19
⊢ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ V |
114 | | eqid 2772 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
115 | 114 | fvmpt2 6603 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑥 ∈ 𝐴 ∧ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ V) → ((𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})‘𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
116 | 113, 115 | mpan2 678 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈ 𝐴 → ((𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})‘𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
117 | 112, 116 | sylan9eq 2828 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) = {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) |
118 | 117 | eleq2d 2845 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) ↔ 𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
119 | | rabid 3311 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 ∈ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ↔ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦)) |
120 | 118, 119 | syl6bb 279 |
. . . . . . . . . . . . . . 15
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ (𝑓‘𝑥) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦))) |
121 | 109, 120 | syldan 582 |
. . . . . . . . . . . . . 14
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑦 ∈ (𝑓‘𝑥) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥𝑟𝑦))) |
122 | 110, 111,
121 | mpbir2and 700 |
. . . . . . . . . . . . 13
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → 𝑦 ∈ (𝑓‘𝑥)) |
123 | 109, 122 | jca 504 |
. . . . . . . . . . . 12
⊢ (((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥𝑟𝑦) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))) |
124 | 123 | ex 405 |
. . . . . . . . . . 11
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑥𝑟𝑦 → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
125 | 120 | simplbda 492 |
. . . . . . . . . . . 12
⊢ ((((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑥𝑟𝑦) |
126 | 125 | expl 450 |
. . . . . . . . . . 11
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) → 𝑥𝑟𝑦)) |
127 | 124, 126 | impbid 204 |
. . . . . . . . . 10
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑥𝑟𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
128 | 45, 127 | syl5bbr 277 |
. . . . . . . . 9
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (〈𝑥, 𝑦〉 ∈ 𝑟 ↔ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)))) |
129 | 128, 47 | syl6bbr 281 |
. . . . . . . 8
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (〈𝑥, 𝑦〉 ∈ 𝑟 ↔ 〈𝑥, 𝑦〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})) |
130 | 97, 103, 104, 105, 33, 39, 129 | eqrelrd2 30145 |
. . . . . . 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))}) |
131 | 90, 92, 93, 130 | syl21anc 825 |
. . . . . 6
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) |
132 | 84, 131 | jca 504 |
. . . . 5
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝑓 ∈ (𝒫 𝐵 ↑𝑚 𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})) |
133 | 72, 132 | impbii 201 |
. . . 4
⊢ ((𝑓 ∈ (𝒫 𝐵 ↑𝑚
𝐴) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) |
134 | 133 | 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})))) |
135 | 1, 18, 20, 134 | f1od 7213 |
. 2
⊢ (⊤
→ 𝑀:(𝒫 𝐵 ↑𝑚
𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵)) |
136 | 135 | mptru 1514 |
1
⊢ 𝑀:(𝒫 𝐵 ↑𝑚 𝐴)–1-1-onto→𝒫 (𝐴 × 𝐵) |