Step | Hyp | Ref
| Expression |
1 | | eqid 2739 |
. . 3
⊢ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
2 | | rfovd.b |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈ 𝑊) |
3 | | ssrab2 4017 |
. . . . . . . . 9
⊢ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ⊆ 𝐵 |
4 | 3 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ⊆ 𝐵) |
5 | 2, 4 | sselpwd 5253 |
. . . . . . 7
⊢ (𝜑 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ 𝒫 𝐵) |
6 | 5 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ 𝒫 𝐵) |
7 | 6 | fmpttd 6983 |
. . . . 5
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵) |
8 | 2 | pwexd 5305 |
. . . . . 6
⊢ (𝜑 → 𝒫 𝐵 ∈ V) |
9 | | rfovd.a |
. . . . . 6
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
10 | 8, 9 | elmapd 8603 |
. . . . 5
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) ∈ (𝒫 𝐵 ↑m 𝐴) ↔ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}):𝐴⟶𝒫 𝐵)) |
11 | 7, 10 | mpbird 256 |
. . . 4
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) ∈ (𝒫 𝐵 ↑m 𝐴)) |
12 | 11 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) ∈ (𝒫 𝐵 ↑m 𝐴)) |
13 | 9, 2 | xpexd 7592 |
. . . . 5
⊢ (𝜑 → (𝐴 × 𝐵) ∈ V) |
14 | 13 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → (𝐴 × 𝐵) ∈ V) |
15 | 8, 9 | elmapd 8603 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝒫 𝐵)) |
16 | 15 | biimpa 476 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → 𝑓:𝐴⟶𝒫 𝐵) |
17 | 16 | ffvelrnda 6955 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) ∧ 𝑥 ∈ 𝐴) → (𝑓‘𝑥) ∈ 𝒫 𝐵) |
18 | 17 | ex 412 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → (𝑥 ∈ 𝐴 → (𝑓‘𝑥) ∈ 𝒫 𝐵)) |
19 | | elpwi 4547 |
. . . . . . . . . 10
⊢ ((𝑓‘𝑥) ∈ 𝒫 𝐵 → (𝑓‘𝑥) ⊆ 𝐵) |
20 | 19 | sseld 3924 |
. . . . . . . . 9
⊢ ((𝑓‘𝑥) ∈ 𝒫 𝐵 → (𝑦 ∈ (𝑓‘𝑥) → 𝑦 ∈ 𝐵)) |
21 | 18, 20 | syl6 35 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → (𝑥 ∈ 𝐴 → (𝑦 ∈ (𝑓‘𝑥) → 𝑦 ∈ 𝐵))) |
22 | 21 | imdistand 570 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) → (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))) |
23 | | trud 1551 |
. . . . . . 7
⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) → ⊤) |
24 | 22, 23 | jca2 513 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⊤))) |
25 | 24 | ssopab2dv 5465 |
. . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ⊆ {〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⊤)}) |
26 | | opabssxp 5677 |
. . . . 5
⊢
{〈𝑥, 𝑦〉 ∣ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ ⊤)} ⊆ (𝐴 × 𝐵) |
27 | 25, 26 | sstrdi 3937 |
. . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ⊆ (𝐴 × 𝐵)) |
28 | 14, 27 | sselpwd 5253 |
. . 3
⊢ ((𝜑 ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ∈ 𝒫 (𝐴 × 𝐵)) |
29 | | simplrr 774 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) |
30 | | elmapfn 8627 |
. . . . . 6
⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) → 𝑓 Fn 𝐴) |
31 | 29, 30 | syl 17 |
. . . . 5
⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑓 Fn 𝐴) |
32 | 2 | ad2antrr 722 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝐵 ∈ 𝑊) |
33 | | rabexg 5258 |
. . . . . . 7
⊢ (𝐵 ∈ 𝑊 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ V) |
34 | 33 | ralrimivw 3110 |
. . . . . 6
⊢ (𝐵 ∈ 𝑊 → ∀𝑥 ∈ 𝐴 {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ V) |
35 | | nfcv 2908 |
. . . . . . 7
⊢
Ⅎ𝑥𝐴 |
36 | 35 | fnmptf 6565 |
. . . . . 6
⊢
(∀𝑥 ∈
𝐴 {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} ∈ V → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) Fn 𝐴) |
37 | 32, 34, 36 | 3syl 18 |
. . . . 5
⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) Fn 𝐴) |
38 | | dfin5 3899 |
. . . . . . 7
⊢ (𝐵 ∩ (𝑓‘𝑢)) = {𝑏 ∈ 𝐵 ∣ 𝑏 ∈ (𝑓‘𝑢)} |
39 | | simpllr 772 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) |
40 | | elmapi 8611 |
. . . . . . . . . . 11
⊢ (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) → 𝑓:𝐴⟶𝒫 𝐵) |
41 | 39, 40 | simpl2im 503 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → 𝑓:𝐴⟶𝒫 𝐵) |
42 | | simpr 484 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴) |
43 | 41, 42 | ffvelrnd 6956 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) ∈ 𝒫 𝐵) |
44 | 43 | elpwid 4549 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) ⊆ 𝐵) |
45 | | sseqin2 4154 |
. . . . . . . 8
⊢ ((𝑓‘𝑢) ⊆ 𝐵 ↔ (𝐵 ∩ (𝑓‘𝑢)) = (𝑓‘𝑢)) |
46 | 44, 45 | sylib 217 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → (𝐵 ∩ (𝑓‘𝑢)) = (𝑓‘𝑢)) |
47 | | ibar 528 |
. . . . . . . . 9
⊢ (𝑢 ∈ 𝐴 → (𝑏 ∈ (𝑓‘𝑢) ↔ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢)))) |
48 | 47 | rabbidv 3412 |
. . . . . . . 8
⊢ (𝑢 ∈ 𝐴 → {𝑏 ∈ 𝐵 ∣ 𝑏 ∈ (𝑓‘𝑢)} = {𝑏 ∈ 𝐵 ∣ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))}) |
49 | 48 | adantl 481 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → {𝑏 ∈ 𝐵 ∣ 𝑏 ∈ (𝑓‘𝑢)} = {𝑏 ∈ 𝐵 ∣ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))}) |
50 | 38, 46, 49 | 3eqtr3a 2803 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) = {𝑏 ∈ 𝐵 ∣ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))}) |
51 | | breq2 5082 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑏 → (𝑥𝑟𝑦 ↔ 𝑥𝑟𝑏)) |
52 | 51 | cbvrabv 3424 |
. . . . . . . . 9
⊢ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} = {𝑏 ∈ 𝐵 ∣ 𝑥𝑟𝑏} |
53 | | breq1 5081 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑎 → (𝑥𝑟𝑏 ↔ 𝑎𝑟𝑏)) |
54 | | df-br 5079 |
. . . . . . . . . . 11
⊢ (𝑎𝑟𝑏 ↔ 〈𝑎, 𝑏〉 ∈ 𝑟) |
55 | 53, 54 | bitrdi 286 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑎 → (𝑥𝑟𝑏 ↔ 〈𝑎, 𝑏〉 ∈ 𝑟)) |
56 | 55 | rabbidv 3412 |
. . . . . . . . 9
⊢ (𝑥 = 𝑎 → {𝑏 ∈ 𝐵 ∣ 𝑥𝑟𝑏} = {𝑏 ∈ 𝐵 ∣ 〈𝑎, 𝑏〉 ∈ 𝑟}) |
57 | 52, 56 | eqtrid 2791 |
. . . . . . . 8
⊢ (𝑥 = 𝑎 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} = {𝑏 ∈ 𝐵 ∣ 〈𝑎, 𝑏〉 ∈ 𝑟}) |
58 | 57 | cbvmptv 5191 |
. . . . . . 7
⊢ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) = (𝑎 ∈ 𝐴 ↦ {𝑏 ∈ 𝐵 ∣ 〈𝑎, 𝑏〉 ∈ 𝑟}) |
59 | | simpr 484 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑎 = 𝑢) → 𝑎 = 𝑢) |
60 | 59 | opeq1d 4815 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑎 = 𝑢) → 〈𝑎, 𝑏〉 = 〈𝑢, 𝑏〉) |
61 | | simpllr 772 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑎 = 𝑢) → 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) |
62 | 60, 61 | eleq12d 2834 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑎 = 𝑢) → (〈𝑎, 𝑏〉 ∈ 𝑟 ↔ 〈𝑢, 𝑏〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})) |
63 | | vex 3434 |
. . . . . . . . . 10
⊢ 𝑢 ∈ V |
64 | | vex 3434 |
. . . . . . . . . 10
⊢ 𝑏 ∈ V |
65 | | simpl 482 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑏) → 𝑥 = 𝑢) |
66 | 65 | eleq1d 2824 |
. . . . . . . . . . 11
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑏) → (𝑥 ∈ 𝐴 ↔ 𝑢 ∈ 𝐴)) |
67 | | simpr 484 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑏) → 𝑦 = 𝑏) |
68 | 65 | fveq2d 6772 |
. . . . . . . . . . . 12
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑏) → (𝑓‘𝑥) = (𝑓‘𝑢)) |
69 | 67, 68 | eleq12d 2834 |
. . . . . . . . . . 11
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑏) → (𝑦 ∈ (𝑓‘𝑥) ↔ 𝑏 ∈ (𝑓‘𝑢))) |
70 | 66, 69 | anbi12d 630 |
. . . . . . . . . 10
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑏) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) ↔ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢)))) |
71 | 63, 64, 70 | opelopaba 5450 |
. . . . . . . . 9
⊢
(〈𝑢, 𝑏〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ↔ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))) |
72 | 62, 71 | bitrdi 286 |
. . . . . . . 8
⊢
(((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑎 = 𝑢) → (〈𝑎, 𝑏〉 ∈ 𝑟 ↔ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢)))) |
73 | 72 | rabbidv 3412 |
. . . . . . 7
⊢
(((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) ∧ 𝑎 = 𝑢) → {𝑏 ∈ 𝐵 ∣ 〈𝑎, 𝑏〉 ∈ 𝑟} = {𝑏 ∈ 𝐵 ∣ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))}) |
74 | 2 | ad3antrrr 726 |
. . . . . . . 8
⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → 𝐵 ∈ 𝑊) |
75 | | rabexg 5258 |
. . . . . . . 8
⊢ (𝐵 ∈ 𝑊 → {𝑏 ∈ 𝐵 ∣ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))} ∈ V) |
76 | 74, 75 | syl 17 |
. . . . . . 7
⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → {𝑏 ∈ 𝐵 ∣ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))} ∈ V) |
77 | 58, 73, 42, 76 | fvmptd2 6877 |
. . . . . 6
⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})‘𝑢) = {𝑏 ∈ 𝐵 ∣ (𝑢 ∈ 𝐴 ∧ 𝑏 ∈ (𝑓‘𝑢))}) |
78 | 50, 77 | eqtr4d 2782 |
. . . . 5
⊢ ((((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) = ((𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})‘𝑢)) |
79 | 31, 37, 78 | eqfnfvd 6906 |
. . . 4
⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) → 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
80 | | simplrl 773 |
. . . . . . . 8
⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) |
81 | 80 | elpwid 4549 |
. . . . . . 7
⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵)) |
82 | | xpss 5604 |
. . . . . . 7
⊢ (𝐴 × 𝐵) ⊆ (V × V) |
83 | 81, 82 | sstrdi 3937 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ⊆ (V × V)) |
84 | | df-rel 5595 |
. . . . . 6
⊢ (Rel
𝑟 ↔ 𝑟 ⊆ (V × V)) |
85 | 83, 84 | sylibr 233 |
. . . . 5
⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → Rel 𝑟) |
86 | | relopabv 5728 |
. . . . . 6
⊢ Rel
{〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} |
87 | 86 | a1i 11 |
. . . . 5
⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → Rel {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) |
88 | | simpl 482 |
. . . . . . 7
⊢ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴)) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) |
89 | 2, 88 | anim12i 612 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) → (𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵))) |
90 | 89 | anim1i 614 |
. . . . 5
⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → ((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) |
91 | | vex 3434 |
. . . . . . . 8
⊢ 𝑣 ∈ V |
92 | | simpl 482 |
. . . . . . . . . 10
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑥 = 𝑢) |
93 | 92 | eleq1d 2824 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑥 ∈ 𝐴 ↔ 𝑢 ∈ 𝐴)) |
94 | | simpr 484 |
. . . . . . . . . 10
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → 𝑦 = 𝑣) |
95 | 92 | fveq2d 6772 |
. . . . . . . . . 10
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑓‘𝑥) = (𝑓‘𝑢)) |
96 | 94, 95 | eleq12d 2834 |
. . . . . . . . 9
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → (𝑦 ∈ (𝑓‘𝑥) ↔ 𝑣 ∈ (𝑓‘𝑢))) |
97 | 93, 96 | anbi12d 630 |
. . . . . . . 8
⊢ ((𝑥 = 𝑢 ∧ 𝑦 = 𝑣) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥)) ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢)))) |
98 | 63, 91, 97 | opelopaba 5450 |
. . . . . . 7
⊢
(〈𝑢, 𝑣〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢))) |
99 | | breq2 5082 |
. . . . . . . . . . . 12
⊢ (𝑏 = 𝑣 → (𝑢𝑟𝑏 ↔ 𝑢𝑟𝑣)) |
100 | | df-br 5079 |
. . . . . . . . . . . 12
⊢ (𝑢𝑟𝑣 ↔ 〈𝑢, 𝑣〉 ∈ 𝑟) |
101 | 99, 100 | bitrdi 286 |
. . . . . . . . . . 11
⊢ (𝑏 = 𝑣 → (𝑢𝑟𝑏 ↔ 〈𝑢, 𝑣〉 ∈ 𝑟)) |
102 | 101 | elrab 3625 |
. . . . . . . . . 10
⊢ (𝑣 ∈ {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏} ↔ (𝑣 ∈ 𝐵 ∧ 〈𝑢, 𝑣〉 ∈ 𝑟)) |
103 | 102 | anbi2i 622 |
. . . . . . . . 9
⊢ ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏}) ↔ (𝑢 ∈ 𝐴 ∧ (𝑣 ∈ 𝐵 ∧ 〈𝑢, 𝑣〉 ∈ 𝑟))) |
104 | 103 | a1i 11 |
. . . . . . . 8
⊢ (((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏}) ↔ (𝑢 ∈ 𝐴 ∧ (𝑣 ∈ 𝐵 ∧ 〈𝑢, 𝑣〉 ∈ 𝑟)))) |
105 | | simplr 765 |
. . . . . . . . . . . 12
⊢ ((((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑢 ∈ 𝐴) → 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) |
106 | | breq1 5081 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑎 → (𝑥𝑟𝑦 ↔ 𝑎𝑟𝑦)) |
107 | 106 | rabbidv 3412 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑎 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} = {𝑦 ∈ 𝐵 ∣ 𝑎𝑟𝑦}) |
108 | | breq2 5082 |
. . . . . . . . . . . . . . 15
⊢ (𝑦 = 𝑏 → (𝑎𝑟𝑦 ↔ 𝑎𝑟𝑏)) |
109 | 108 | cbvrabv 3424 |
. . . . . . . . . . . . . 14
⊢ {𝑦 ∈ 𝐵 ∣ 𝑎𝑟𝑦} = {𝑏 ∈ 𝐵 ∣ 𝑎𝑟𝑏} |
110 | 107, 109 | eqtrdi 2795 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑎 → {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦} = {𝑏 ∈ 𝐵 ∣ 𝑎𝑟𝑏}) |
111 | 110 | cbvmptv 5191 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}) = (𝑎 ∈ 𝐴 ↦ {𝑏 ∈ 𝐵 ∣ 𝑎𝑟𝑏}) |
112 | 105, 111 | eqtrdi 2795 |
. . . . . . . . . . 11
⊢ ((((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑢 ∈ 𝐴) → 𝑓 = (𝑎 ∈ 𝐴 ↦ {𝑏 ∈ 𝐵 ∣ 𝑎𝑟𝑏})) |
113 | | breq1 5081 |
. . . . . . . . . . . . 13
⊢ (𝑎 = 𝑢 → (𝑎𝑟𝑏 ↔ 𝑢𝑟𝑏)) |
114 | 113 | rabbidv 3412 |
. . . . . . . . . . . 12
⊢ (𝑎 = 𝑢 → {𝑏 ∈ 𝐵 ∣ 𝑎𝑟𝑏} = {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏}) |
115 | 114 | adantl 481 |
. . . . . . . . . . 11
⊢
(((((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑢 ∈ 𝐴) ∧ 𝑎 = 𝑢) → {𝑏 ∈ 𝐵 ∣ 𝑎𝑟𝑏} = {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏}) |
116 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑢 ∈ 𝐴) → 𝑢 ∈ 𝐴) |
117 | | rabexg 5258 |
. . . . . . . . . . . 12
⊢ (𝐵 ∈ 𝑊 → {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏} ∈ V) |
118 | 117 | ad3antrrr 726 |
. . . . . . . . . . 11
⊢ ((((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑢 ∈ 𝐴) → {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏} ∈ V) |
119 | 112, 115,
116, 118 | fvmptd 6876 |
. . . . . . . . . 10
⊢ ((((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑢 ∈ 𝐴) → (𝑓‘𝑢) = {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏}) |
120 | 119 | eleq2d 2825 |
. . . . . . . . 9
⊢ ((((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) ∧ 𝑢 ∈ 𝐴) → (𝑣 ∈ (𝑓‘𝑢) ↔ 𝑣 ∈ {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏})) |
121 | 120 | pm5.32da 578 |
. . . . . . . 8
⊢ (((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢)) ↔ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ {𝑏 ∈ 𝐵 ∣ 𝑢𝑟𝑏}))) |
122 | | simplr 765 |
. . . . . . . . . 10
⊢ (((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) |
123 | 122 | elpwid 4549 |
. . . . . . . . 9
⊢ (((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 ⊆ (𝐴 × 𝐵)) |
124 | 63, 91 | opeldm 5813 |
. . . . . . . . . . . 12
⊢
(〈𝑢, 𝑣〉 ∈ 𝑟 → 𝑢 ∈ dom 𝑟) |
125 | | dmss 5808 |
. . . . . . . . . . . . . 14
⊢ (𝑟 ⊆ (𝐴 × 𝐵) → dom 𝑟 ⊆ dom (𝐴 × 𝐵)) |
126 | | dmxpss 6071 |
. . . . . . . . . . . . . 14
⊢ dom
(𝐴 × 𝐵) ⊆ 𝐴 |
127 | 125, 126 | sstrdi 3937 |
. . . . . . . . . . . . 13
⊢ (𝑟 ⊆ (𝐴 × 𝐵) → dom 𝑟 ⊆ 𝐴) |
128 | 127 | sseld 3924 |
. . . . . . . . . . . 12
⊢ (𝑟 ⊆ (𝐴 × 𝐵) → (𝑢 ∈ dom 𝑟 → 𝑢 ∈ 𝐴)) |
129 | 124, 128 | syl5 34 |
. . . . . . . . . . 11
⊢ (𝑟 ⊆ (𝐴 × 𝐵) → (〈𝑢, 𝑣〉 ∈ 𝑟 → 𝑢 ∈ 𝐴)) |
130 | 129 | pm4.71rd 562 |
. . . . . . . . . 10
⊢ (𝑟 ⊆ (𝐴 × 𝐵) → (〈𝑢, 𝑣〉 ∈ 𝑟 ↔ (𝑢 ∈ 𝐴 ∧ 〈𝑢, 𝑣〉 ∈ 𝑟))) |
131 | 63, 91 | opelrn 5849 |
. . . . . . . . . . . . 13
⊢
(〈𝑢, 𝑣〉 ∈ 𝑟 → 𝑣 ∈ ran 𝑟) |
132 | | rnss 5845 |
. . . . . . . . . . . . . . 15
⊢ (𝑟 ⊆ (𝐴 × 𝐵) → ran 𝑟 ⊆ ran (𝐴 × 𝐵)) |
133 | | rnxpss 6072 |
. . . . . . . . . . . . . . 15
⊢ ran
(𝐴 × 𝐵) ⊆ 𝐵 |
134 | 132, 133 | sstrdi 3937 |
. . . . . . . . . . . . . 14
⊢ (𝑟 ⊆ (𝐴 × 𝐵) → ran 𝑟 ⊆ 𝐵) |
135 | 134 | sseld 3924 |
. . . . . . . . . . . . 13
⊢ (𝑟 ⊆ (𝐴 × 𝐵) → (𝑣 ∈ ran 𝑟 → 𝑣 ∈ 𝐵)) |
136 | 131, 135 | syl5 34 |
. . . . . . . . . . . 12
⊢ (𝑟 ⊆ (𝐴 × 𝐵) → (〈𝑢, 𝑣〉 ∈ 𝑟 → 𝑣 ∈ 𝐵)) |
137 | 136 | pm4.71rd 562 |
. . . . . . . . . . 11
⊢ (𝑟 ⊆ (𝐴 × 𝐵) → (〈𝑢, 𝑣〉 ∈ 𝑟 ↔ (𝑣 ∈ 𝐵 ∧ 〈𝑢, 𝑣〉 ∈ 𝑟))) |
138 | 137 | anbi2d 628 |
. . . . . . . . . 10
⊢ (𝑟 ⊆ (𝐴 × 𝐵) → ((𝑢 ∈ 𝐴 ∧ 〈𝑢, 𝑣〉 ∈ 𝑟) ↔ (𝑢 ∈ 𝐴 ∧ (𝑣 ∈ 𝐵 ∧ 〈𝑢, 𝑣〉 ∈ 𝑟)))) |
139 | 130, 138 | bitrd 278 |
. . . . . . . . 9
⊢ (𝑟 ⊆ (𝐴 × 𝐵) → (〈𝑢, 𝑣〉 ∈ 𝑟 ↔ (𝑢 ∈ 𝐴 ∧ (𝑣 ∈ 𝐵 ∧ 〈𝑢, 𝑣〉 ∈ 𝑟)))) |
140 | 123, 139 | syl 17 |
. . . . . . . 8
⊢ (((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (〈𝑢, 𝑣〉 ∈ 𝑟 ↔ (𝑢 ∈ 𝐴 ∧ (𝑣 ∈ 𝐵 ∧ 〈𝑢, 𝑣〉 ∈ 𝑟)))) |
141 | 104, 121,
140 | 3bitr4d 310 |
. . . . . . 7
⊢ (((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → ((𝑢 ∈ 𝐴 ∧ 𝑣 ∈ (𝑓‘𝑢)) ↔ 〈𝑢, 𝑣〉 ∈ 𝑟)) |
142 | 98, 141 | bitr2id 283 |
. . . . . 6
⊢ (((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (〈𝑢, 𝑣〉 ∈ 𝑟 ↔ 〈𝑢, 𝑣〉 ∈ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})) |
143 | 142 | eqrelrdv2 5702 |
. . . . 5
⊢ (((Rel
𝑟 ∧ Rel {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ∧ ((𝐵 ∈ 𝑊 ∧ 𝑟 ∈ 𝒫 (𝐴 × 𝐵)) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) → 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) |
144 | 85, 87, 90, 143 | syl21anc 834 |
. . . 4
⊢ (((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) ∧ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → 𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) |
145 | 79, 144 | impbida 797 |
. . 3
⊢ ((𝜑 ∧ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ∧ 𝑓 ∈ (𝒫 𝐵 ↑m 𝐴))) → (𝑟 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))} ↔ 𝑓 = (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) |
146 | 1, 12, 28, 145 | f1ocnv2d 7513 |
. 2
⊢ (𝜑 → ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})):𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝐴) ∧ ◡(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))) |
147 | | rfovcnvf1od.f |
. . . 4
⊢ 𝐹 = (𝐴𝑂𝐵) |
148 | | rfovd.rf |
. . . . 5
⊢ 𝑂 = (𝑎 ∈ V, 𝑏 ∈ V ↦ (𝑟 ∈ 𝒫 (𝑎 × 𝑏) ↦ (𝑥 ∈ 𝑎 ↦ {𝑦 ∈ 𝑏 ∣ 𝑥𝑟𝑦}))) |
149 | 148, 9, 2 | rfovd 41562 |
. . . 4
⊢ (𝜑 → (𝐴𝑂𝐵) = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) |
150 | 147, 149 | eqtrid 2791 |
. . 3
⊢ (𝜑 → 𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) |
151 | | f1oeq1 6700 |
. . . 4
⊢ (𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝐴) ↔ (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})):𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝐴))) |
152 | | cnveq 5779 |
. . . . 5
⊢ (𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → ◡𝐹 = ◡(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦}))) |
153 | 152 | eqeq1d 2741 |
. . . 4
⊢ (𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → (◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}) ↔ ◡(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))) |
154 | 151, 153 | anbi12d 630 |
. . 3
⊢ (𝐹 = (𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) → ((𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝐴) ∧ ◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})) ↔ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})):𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝐴) ∧ ◡(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})))) |
155 | 150, 154 | syl 17 |
. 2
⊢ (𝜑 → ((𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝐴) ∧ ◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})) ↔ ((𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})):𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝐴) ∧ ◡(𝑟 ∈ 𝒫 (𝐴 × 𝐵) ↦ (𝑥 ∈ 𝐴 ↦ {𝑦 ∈ 𝐵 ∣ 𝑥𝑟𝑦})) = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))})))) |
156 | 146, 155 | mpbird 256 |
1
⊢ (𝜑 → (𝐹:𝒫 (𝐴 × 𝐵)–1-1-onto→(𝒫 𝐵 ↑m 𝐴) ∧ ◡𝐹 = (𝑓 ∈ (𝒫 𝐵 ↑m 𝐴) ↦ {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ (𝑓‘𝑥))}))) |