Step | Hyp | Ref
| Expression |
1 | | opex 4932 |
. . . . . . . . 9
⊢
〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ V |
2 | | fvex 6201 |
. . . . . . . . 9
⊢ (𝐹‘(𝑝 · 𝑞)) ∈ V |
3 | 1, 2 | relsnop 5224 |
. . . . . . . 8
⊢ Rel
{〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} |
4 | 3 | rgenw 2924 |
. . . . . . 7
⊢
∀𝑞 ∈
𝑉 Rel {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} |
5 | | reliun 5239 |
. . . . . . 7
⊢ (Rel
∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ↔ ∀𝑞 ∈ 𝑉 Rel {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
6 | 4, 5 | mpbir 221 |
. . . . . 6
⊢ Rel
∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} |
7 | 6 | rgenw 2924 |
. . . . 5
⊢
∀𝑝 ∈
𝑉 Rel ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} |
8 | | reliun 5239 |
. . . . 5
⊢ (Rel
∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ↔ ∀𝑝 ∈ 𝑉 Rel ∪
𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
9 | 7, 8 | mpbir 221 |
. . . 4
⊢ Rel
∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} |
10 | | imasaddflem.a |
. . . . 5
⊢ (𝜑 → ∙ = ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
11 | 10 | releqd 5203 |
. . . 4
⊢ (𝜑 → (Rel ∙ ↔ Rel ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉})) |
12 | 9, 11 | mpbiri 248 |
. . 3
⊢ (𝜑 → Rel ∙ ) |
13 | | imasaddf.f |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐹:𝑉–onto→𝐵) |
14 | | fof 6115 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐹:𝑉–onto→𝐵 → 𝐹:𝑉⟶𝐵) |
15 | 13, 14 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝐹:𝑉⟶𝐵) |
16 | | ffvelrn 6357 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹:𝑉⟶𝐵 ∧ 𝑝 ∈ 𝑉) → (𝐹‘𝑝) ∈ 𝐵) |
17 | | ffvelrn 6357 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹:𝑉⟶𝐵 ∧ 𝑞 ∈ 𝑉) → (𝐹‘𝑞) ∈ 𝐵) |
18 | 16, 17 | anim12dan 882 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹:𝑉⟶𝐵 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((𝐹‘𝑝) ∈ 𝐵 ∧ (𝐹‘𝑞) ∈ 𝐵)) |
19 | 15, 18 | sylan 488 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((𝐹‘𝑝) ∈ 𝐵 ∧ (𝐹‘𝑞) ∈ 𝐵)) |
20 | | opelxpi 5148 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐹‘𝑝) ∈ 𝐵 ∧ (𝐹‘𝑞) ∈ 𝐵) → 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ (𝐵 × 𝐵)) |
21 | 19, 20 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ (𝐵 × 𝐵)) |
22 | | opelxpi 5148 |
. . . . . . . . . . . . . . 15
⊢
((〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ (𝐵 × 𝐵) ∧ (𝐹‘(𝑝 · 𝑞)) ∈ V) → 〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉 ∈ ((𝐵 × 𝐵) × V)) |
23 | 21, 2, 22 | sylancl 694 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → 〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉 ∈ ((𝐵 × 𝐵) × V)) |
24 | 23 | snssd 4340 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ((𝐵 × 𝐵) × V)) |
25 | 24 | anassrs 680 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑝 ∈ 𝑉) ∧ 𝑞 ∈ 𝑉) → {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ((𝐵 × 𝐵) × V)) |
26 | 25 | ralrimiva 2966 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑝 ∈ 𝑉) → ∀𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ((𝐵 × 𝐵) × V)) |
27 | | iunss 4561 |
. . . . . . . . . . 11
⊢ (∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ((𝐵 × 𝐵) × V) ↔ ∀𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ((𝐵 × 𝐵) × V)) |
28 | 26, 27 | sylibr 224 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑝 ∈ 𝑉) → ∪
𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ((𝐵 × 𝐵) × V)) |
29 | 28 | ralrimiva 2966 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ((𝐵 × 𝐵) × V)) |
30 | | iunss 4561 |
. . . . . . . . 9
⊢ (∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ((𝐵 × 𝐵) × V) ↔ ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ((𝐵 × 𝐵) × V)) |
31 | 29, 30 | sylibr 224 |
. . . . . . . 8
⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ((𝐵 × 𝐵) × V)) |
32 | 10, 31 | eqsstrd 3639 |
. . . . . . 7
⊢ (𝜑 → ∙ ⊆ ((𝐵 × 𝐵) × V)) |
33 | | dmss 5323 |
. . . . . . 7
⊢ ( ∙
⊆ ((𝐵 × 𝐵) × V) → dom ∙
⊆ dom ((𝐵 ×
𝐵) ×
V)) |
34 | 32, 33 | syl 17 |
. . . . . 6
⊢ (𝜑 → dom ∙ ⊆ dom ((𝐵 × 𝐵) × V)) |
35 | | vn0 3924 |
. . . . . . 7
⊢ V ≠
∅ |
36 | | dmxp 5344 |
. . . . . . 7
⊢ (V ≠
∅ → dom ((𝐵
× 𝐵) × V) =
(𝐵 × 𝐵)) |
37 | 35, 36 | ax-mp 5 |
. . . . . 6
⊢ dom
((𝐵 × 𝐵) × V) = (𝐵 × 𝐵) |
38 | 34, 37 | syl6sseq 3651 |
. . . . 5
⊢ (𝜑 → dom ∙ ⊆ (𝐵 × 𝐵)) |
39 | | forn 6118 |
. . . . . . 7
⊢ (𝐹:𝑉–onto→𝐵 → ran 𝐹 = 𝐵) |
40 | 13, 39 | syl 17 |
. . . . . 6
⊢ (𝜑 → ran 𝐹 = 𝐵) |
41 | 40 | sqxpeqd 5141 |
. . . . 5
⊢ (𝜑 → (ran 𝐹 × ran 𝐹) = (𝐵 × 𝐵)) |
42 | 38, 41 | sseqtr4d 3642 |
. . . 4
⊢ (𝜑 → dom ∙ ⊆ (ran 𝐹 × ran 𝐹)) |
43 | 10 | eleq2d 2687 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ ∙ ↔
〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉})) |
44 | 43 | adantr 481 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ ∙ ↔
〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉})) |
45 | | df-br 4654 |
. . . . . . . . . . . 12
⊢
(〈(𝐹‘𝑎), (𝐹‘𝑏)〉 ∙ 𝑤 ↔ 〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ ∙ ) |
46 | | eliun 4524 |
. . . . . . . . . . . . 13
⊢
(〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ↔ ∃𝑝 ∈ 𝑉 〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
47 | | eliun 4524 |
. . . . . . . . . . . . . 14
⊢
(〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ↔ ∃𝑞 ∈ 𝑉 〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
48 | 47 | rexbii 3041 |
. . . . . . . . . . . . 13
⊢
(∃𝑝 ∈
𝑉 〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ↔ ∃𝑝 ∈ 𝑉 ∃𝑞 ∈ 𝑉 〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
49 | 46, 48 | bitr2i 265 |
. . . . . . . . . . . 12
⊢
(∃𝑝 ∈
𝑉 ∃𝑞 ∈ 𝑉 〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ↔ 〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉}) |
50 | 44, 45, 49 | 3bitr4g 303 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (〈(𝐹‘𝑎), (𝐹‘𝑏)〉 ∙ 𝑤 ↔ ∃𝑝 ∈ 𝑉 ∃𝑞 ∈ 𝑉 〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉})) |
51 | | opex 4932 |
. . . . . . . . . . . . . . 15
⊢
〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ V |
52 | 51 | elsn 4192 |
. . . . . . . . . . . . . 14
⊢
(〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ↔ 〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 = 〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉) |
53 | | opex 4932 |
. . . . . . . . . . . . . . . 16
⊢
〈(𝐹‘𝑎), (𝐹‘𝑏)〉 ∈ V |
54 | | vex 3203 |
. . . . . . . . . . . . . . . 16
⊢ 𝑤 ∈ V |
55 | 53, 54 | opth 4945 |
. . . . . . . . . . . . . . 15
⊢
(〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 = 〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉 ↔ (〈(𝐹‘𝑎), (𝐹‘𝑏)〉 = 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∧ 𝑤 = (𝐹‘(𝑝 · 𝑞)))) |
56 | | fvex 6201 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐹‘𝑎) ∈ V |
57 | | fvex 6201 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐹‘𝑏) ∈ V |
58 | 56, 57 | opth 4945 |
. . . . . . . . . . . . . . . . . 18
⊢
(〈(𝐹‘𝑎), (𝐹‘𝑏)〉 = 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ↔ ((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞))) |
59 | | imasaddf.e |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (((𝐹‘𝑎) = (𝐹‘𝑝) ∧ (𝐹‘𝑏) = (𝐹‘𝑞)) → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) |
60 | 58, 59 | syl5bi 232 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (〈(𝐹‘𝑎), (𝐹‘𝑏)〉 = 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 → (𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)))) |
61 | | eqeq2 2633 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)) → (𝑤 = (𝐹‘(𝑎 · 𝑏)) ↔ 𝑤 = (𝐹‘(𝑝 · 𝑞)))) |
62 | 61 | biimprd 238 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹‘(𝑎 · 𝑏)) = (𝐹‘(𝑝 · 𝑞)) → (𝑤 = (𝐹‘(𝑝 · 𝑞)) → 𝑤 = (𝐹‘(𝑎 · 𝑏)))) |
63 | 60, 62 | syl6 35 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (〈(𝐹‘𝑎), (𝐹‘𝑏)〉 = 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 → (𝑤 = (𝐹‘(𝑝 · 𝑞)) → 𝑤 = (𝐹‘(𝑎 · 𝑏))))) |
64 | 63 | impd 447 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → ((〈(𝐹‘𝑎), (𝐹‘𝑏)〉 = 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∧ 𝑤 = (𝐹‘(𝑝 · 𝑞))) → 𝑤 = (𝐹‘(𝑎 · 𝑏)))) |
65 | 55, 64 | syl5bi 232 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 = 〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉 → 𝑤 = (𝐹‘(𝑎 · 𝑏)))) |
66 | 52, 65 | syl5bi 232 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} → 𝑤 = (𝐹‘(𝑎 · 𝑏)))) |
67 | 66 | 3expa 1265 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) ∧ (𝑝 ∈ 𝑉 ∧ 𝑞 ∈ 𝑉)) → (〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} → 𝑤 = (𝐹‘(𝑎 · 𝑏)))) |
68 | 67 | rexlimdvva 3038 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (∃𝑝 ∈ 𝑉 ∃𝑞 ∈ 𝑉 〈〈(𝐹‘𝑎), (𝐹‘𝑏)〉, 𝑤〉 ∈ {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} → 𝑤 = (𝐹‘(𝑎 · 𝑏)))) |
69 | 50, 68 | sylbid 230 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → (〈(𝐹‘𝑎), (𝐹‘𝑏)〉 ∙ 𝑤 → 𝑤 = (𝐹‘(𝑎 · 𝑏)))) |
70 | 69 | alrimiv 1855 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ∀𝑤(〈(𝐹‘𝑎), (𝐹‘𝑏)〉 ∙ 𝑤 → 𝑤 = (𝐹‘(𝑎 · 𝑏)))) |
71 | | mo2icl 3385 |
. . . . . . . . 9
⊢
(∀𝑤(〈(𝐹‘𝑎), (𝐹‘𝑏)〉 ∙ 𝑤 → 𝑤 = (𝐹‘(𝑎 · 𝑏))) → ∃*𝑤〈(𝐹‘𝑎), (𝐹‘𝑏)〉 ∙ 𝑤) |
72 | 70, 71 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉)) → ∃*𝑤〈(𝐹‘𝑎), (𝐹‘𝑏)〉 ∙ 𝑤) |
73 | 72 | ralrimivva 2971 |
. . . . . . 7
⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ∃*𝑤〈(𝐹‘𝑎), (𝐹‘𝑏)〉 ∙ 𝑤) |
74 | | fofn 6117 |
. . . . . . . . . 10
⊢ (𝐹:𝑉–onto→𝐵 → 𝐹 Fn 𝑉) |
75 | 13, 74 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → 𝐹 Fn 𝑉) |
76 | | opeq2 4403 |
. . . . . . . . . . . 12
⊢ (𝑧 = (𝐹‘𝑏) → 〈(𝐹‘𝑎), 𝑧〉 = 〈(𝐹‘𝑎), (𝐹‘𝑏)〉) |
77 | 76 | breq1d 4663 |
. . . . . . . . . . 11
⊢ (𝑧 = (𝐹‘𝑏) → (〈(𝐹‘𝑎), 𝑧〉 ∙ 𝑤 ↔ 〈(𝐹‘𝑎), (𝐹‘𝑏)〉 ∙ 𝑤)) |
78 | 77 | mobidv 2491 |
. . . . . . . . . 10
⊢ (𝑧 = (𝐹‘𝑏) → (∃*𝑤〈(𝐹‘𝑎), 𝑧〉 ∙ 𝑤 ↔ ∃*𝑤〈(𝐹‘𝑎), (𝐹‘𝑏)〉 ∙ 𝑤)) |
79 | 78 | ralrn 6362 |
. . . . . . . . 9
⊢ (𝐹 Fn 𝑉 → (∀𝑧 ∈ ran 𝐹∃*𝑤〈(𝐹‘𝑎), 𝑧〉 ∙ 𝑤 ↔ ∀𝑏 ∈ 𝑉 ∃*𝑤〈(𝐹‘𝑎), (𝐹‘𝑏)〉 ∙ 𝑤)) |
80 | 75, 79 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹∃*𝑤〈(𝐹‘𝑎), 𝑧〉 ∙ 𝑤 ↔ ∀𝑏 ∈ 𝑉 ∃*𝑤〈(𝐹‘𝑎), (𝐹‘𝑏)〉 ∙ 𝑤)) |
81 | 80 | ralbidv 2986 |
. . . . . . 7
⊢ (𝜑 → (∀𝑎 ∈ 𝑉 ∀𝑧 ∈ ran 𝐹∃*𝑤〈(𝐹‘𝑎), 𝑧〉 ∙ 𝑤 ↔ ∀𝑎 ∈ 𝑉 ∀𝑏 ∈ 𝑉 ∃*𝑤〈(𝐹‘𝑎), (𝐹‘𝑏)〉 ∙ 𝑤)) |
82 | 73, 81 | mpbird 247 |
. . . . . 6
⊢ (𝜑 → ∀𝑎 ∈ 𝑉 ∀𝑧 ∈ ran 𝐹∃*𝑤〈(𝐹‘𝑎), 𝑧〉 ∙ 𝑤) |
83 | | opeq1 4402 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝐹‘𝑎) → 〈𝑦, 𝑧〉 = 〈(𝐹‘𝑎), 𝑧〉) |
84 | 83 | breq1d 4663 |
. . . . . . . . . 10
⊢ (𝑦 = (𝐹‘𝑎) → (〈𝑦, 𝑧〉 ∙ 𝑤 ↔ 〈(𝐹‘𝑎), 𝑧〉 ∙ 𝑤)) |
85 | 84 | mobidv 2491 |
. . . . . . . . 9
⊢ (𝑦 = (𝐹‘𝑎) → (∃*𝑤〈𝑦, 𝑧〉 ∙ 𝑤 ↔ ∃*𝑤〈(𝐹‘𝑎), 𝑧〉 ∙ 𝑤)) |
86 | 85 | ralbidv 2986 |
. . . . . . . 8
⊢ (𝑦 = (𝐹‘𝑎) → (∀𝑧 ∈ ran 𝐹∃*𝑤〈𝑦, 𝑧〉 ∙ 𝑤 ↔ ∀𝑧 ∈ ran 𝐹∃*𝑤〈(𝐹‘𝑎), 𝑧〉 ∙ 𝑤)) |
87 | 86 | ralrn 6362 |
. . . . . . 7
⊢ (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐹∃*𝑤〈𝑦, 𝑧〉 ∙ 𝑤 ↔ ∀𝑎 ∈ 𝑉 ∀𝑧 ∈ ran 𝐹∃*𝑤〈(𝐹‘𝑎), 𝑧〉 ∙ 𝑤)) |
88 | 75, 87 | syl 17 |
. . . . . 6
⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐹∃*𝑤〈𝑦, 𝑧〉 ∙ 𝑤 ↔ ∀𝑎 ∈ 𝑉 ∀𝑧 ∈ ran 𝐹∃*𝑤〈(𝐹‘𝑎), 𝑧〉 ∙ 𝑤)) |
89 | 82, 88 | mpbird 247 |
. . . . 5
⊢ (𝜑 → ∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐹∃*𝑤〈𝑦, 𝑧〉 ∙ 𝑤) |
90 | | breq1 4656 |
. . . . . . 7
⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝑥 ∙ 𝑤 ↔ 〈𝑦, 𝑧〉 ∙ 𝑤)) |
91 | 90 | mobidv 2491 |
. . . . . 6
⊢ (𝑥 = 〈𝑦, 𝑧〉 → (∃*𝑤 𝑥 ∙ 𝑤 ↔ ∃*𝑤〈𝑦, 𝑧〉 ∙ 𝑤)) |
92 | 91 | ralxp 5263 |
. . . . 5
⊢
(∀𝑥 ∈
(ran 𝐹 × ran 𝐹)∃*𝑤 𝑥 ∙ 𝑤 ↔ ∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐹∃*𝑤〈𝑦, 𝑧〉 ∙ 𝑤) |
93 | 89, 92 | sylibr 224 |
. . . 4
⊢ (𝜑 → ∀𝑥 ∈ (ran 𝐹 × ran 𝐹)∃*𝑤 𝑥 ∙ 𝑤) |
94 | | ssralv 3666 |
. . . 4
⊢ (dom
∙
⊆ (ran 𝐹 × ran
𝐹) → (∀𝑥 ∈ (ran 𝐹 × ran 𝐹)∃*𝑤 𝑥 ∙ 𝑤 → ∀𝑥 ∈ dom ∙ ∃*𝑤 𝑥 ∙ 𝑤)) |
95 | 42, 93, 94 | sylc 65 |
. . 3
⊢ (𝜑 → ∀𝑥 ∈ dom ∙ ∃*𝑤 𝑥 ∙ 𝑤) |
96 | | dffun7 5915 |
. . 3
⊢ (Fun
∙
↔ (Rel ∙ ∧ ∀𝑥 ∈ dom ∙ ∃*𝑤 𝑥 ∙ 𝑤)) |
97 | 12, 95, 96 | sylanbrc 698 |
. 2
⊢ (𝜑 → Fun ∙ ) |
98 | | eqimss2 3658 |
. . . . . . . . . . 11
⊢ ( ∙ =
∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ∙ ) |
99 | 10, 98 | syl 17 |
. . . . . . . . . 10
⊢ (𝜑 → ∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ∙ ) |
100 | | iunss 4561 |
. . . . . . . . . 10
⊢ (∪ 𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ∙ ↔
∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ∙ ) |
101 | 99, 100 | sylib 208 |
. . . . . . . . 9
⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ∙ ) |
102 | | iunss 4561 |
. . . . . . . . . . 11
⊢ (∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ∙ ↔
∀𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ∙ ) |
103 | | opex 4932 |
. . . . . . . . . . . . . 14
⊢
〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉 ∈ V |
104 | 103 | snss 4316 |
. . . . . . . . . . . . 13
⊢
(〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉 ∈ ∙ ↔
{〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ∙ ) |
105 | 1, 2 | opeldm 5328 |
. . . . . . . . . . . . 13
⊢
(〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉 ∈ ∙ → 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ dom ∙ ) |
106 | 104, 105 | sylbir 225 |
. . . . . . . . . . . 12
⊢
({〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ∙ → 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ dom ∙ ) |
107 | 106 | ralimi 2952 |
. . . . . . . . . . 11
⊢
(∀𝑞 ∈
𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ∙ →
∀𝑞 ∈ 𝑉 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ dom ∙ ) |
108 | 102, 107 | sylbi 207 |
. . . . . . . . . 10
⊢ (∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ∙ →
∀𝑞 ∈ 𝑉 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ dom ∙ ) |
109 | 108 | ralimi 2952 |
. . . . . . . . 9
⊢
(∀𝑝 ∈
𝑉 ∪ 𝑞 ∈ 𝑉 {〈〈(𝐹‘𝑝), (𝐹‘𝑞)〉, (𝐹‘(𝑝 · 𝑞))〉} ⊆ ∙ →
∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑉 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ dom ∙ ) |
110 | 101, 109 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑉 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ dom ∙ ) |
111 | | opeq2 4403 |
. . . . . . . . . . . 12
⊢ (𝑧 = (𝐹‘𝑞) → 〈(𝐹‘𝑝), 𝑧〉 = 〈(𝐹‘𝑝), (𝐹‘𝑞)〉) |
112 | 111 | eleq1d 2686 |
. . . . . . . . . . 11
⊢ (𝑧 = (𝐹‘𝑞) → (〈(𝐹‘𝑝), 𝑧〉 ∈ dom ∙ ↔ 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ dom ∙ )) |
113 | 112 | ralrn 6362 |
. . . . . . . . . 10
⊢ (𝐹 Fn 𝑉 → (∀𝑧 ∈ ran 𝐹〈(𝐹‘𝑝), 𝑧〉 ∈ dom ∙ ↔
∀𝑞 ∈ 𝑉 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ dom ∙ )) |
114 | 75, 113 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (∀𝑧 ∈ ran 𝐹〈(𝐹‘𝑝), 𝑧〉 ∈ dom ∙ ↔
∀𝑞 ∈ 𝑉 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ dom ∙ )) |
115 | 114 | ralbidv 2986 |
. . . . . . . 8
⊢ (𝜑 → (∀𝑝 ∈ 𝑉 ∀𝑧 ∈ ran 𝐹〈(𝐹‘𝑝), 𝑧〉 ∈ dom ∙ ↔
∀𝑝 ∈ 𝑉 ∀𝑞 ∈ 𝑉 〈(𝐹‘𝑝), (𝐹‘𝑞)〉 ∈ dom ∙ )) |
116 | 110, 115 | mpbird 247 |
. . . . . . 7
⊢ (𝜑 → ∀𝑝 ∈ 𝑉 ∀𝑧 ∈ ran 𝐹〈(𝐹‘𝑝), 𝑧〉 ∈ dom ∙ ) |
117 | | opeq1 4402 |
. . . . . . . . . . 11
⊢ (𝑦 = (𝐹‘𝑝) → 〈𝑦, 𝑧〉 = 〈(𝐹‘𝑝), 𝑧〉) |
118 | 117 | eleq1d 2686 |
. . . . . . . . . 10
⊢ (𝑦 = (𝐹‘𝑝) → (〈𝑦, 𝑧〉 ∈ dom ∙ ↔ 〈(𝐹‘𝑝), 𝑧〉 ∈ dom ∙ )) |
119 | 118 | ralbidv 2986 |
. . . . . . . . 9
⊢ (𝑦 = (𝐹‘𝑝) → (∀𝑧 ∈ ran 𝐹〈𝑦, 𝑧〉 ∈ dom ∙ ↔
∀𝑧 ∈ ran 𝐹〈(𝐹‘𝑝), 𝑧〉 ∈ dom ∙ )) |
120 | 119 | ralrn 6362 |
. . . . . . . 8
⊢ (𝐹 Fn 𝑉 → (∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐹〈𝑦, 𝑧〉 ∈ dom ∙ ↔
∀𝑝 ∈ 𝑉 ∀𝑧 ∈ ran 𝐹〈(𝐹‘𝑝), 𝑧〉 ∈ dom ∙ )) |
121 | 75, 120 | syl 17 |
. . . . . . 7
⊢ (𝜑 → (∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐹〈𝑦, 𝑧〉 ∈ dom ∙ ↔
∀𝑝 ∈ 𝑉 ∀𝑧 ∈ ran 𝐹〈(𝐹‘𝑝), 𝑧〉 ∈ dom ∙ )) |
122 | 116, 121 | mpbird 247 |
. . . . . 6
⊢ (𝜑 → ∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐹〈𝑦, 𝑧〉 ∈ dom ∙ ) |
123 | | eleq1 2689 |
. . . . . . 7
⊢ (𝑥 = 〈𝑦, 𝑧〉 → (𝑥 ∈ dom ∙ ↔ 〈𝑦, 𝑧〉 ∈ dom ∙ )) |
124 | 123 | ralxp 5263 |
. . . . . 6
⊢
(∀𝑥 ∈
(ran 𝐹 × ran 𝐹)𝑥 ∈ dom ∙ ↔
∀𝑦 ∈ ran 𝐹∀𝑧 ∈ ran 𝐹〈𝑦, 𝑧〉 ∈ dom ∙ ) |
125 | 122, 124 | sylibr 224 |
. . . . 5
⊢ (𝜑 → ∀𝑥 ∈ (ran 𝐹 × ran 𝐹)𝑥 ∈ dom ∙ ) |
126 | | dfss3 3592 |
. . . . 5
⊢ ((ran
𝐹 × ran 𝐹) ⊆ dom ∙ ↔
∀𝑥 ∈ (ran 𝐹 × ran 𝐹)𝑥 ∈ dom ∙ ) |
127 | 125, 126 | sylibr 224 |
. . . 4
⊢ (𝜑 → (ran 𝐹 × ran 𝐹) ⊆ dom ∙ ) |
128 | 41, 127 | eqsstr3d 3640 |
. . 3
⊢ (𝜑 → (𝐵 × 𝐵) ⊆ dom ∙ ) |
129 | 38, 128 | eqssd 3620 |
. 2
⊢ (𝜑 → dom ∙ = (𝐵 × 𝐵)) |
130 | | df-fn 5891 |
. 2
⊢ ( ∙ Fn
(𝐵 × 𝐵) ↔ (Fun ∙ ∧ dom ∙ =
(𝐵 × 𝐵))) |
131 | 97, 129, 130 | sylanbrc 698 |
1
⊢ (𝜑 → ∙ Fn (𝐵 × 𝐵)) |