Step | Hyp | Ref
| Expression |
1 | | xpcbas.t |
. . . 4
⊢ 𝑇 = (𝐶 ×c 𝐷) |
2 | | xpcbas.x |
. . . 4
⊢ 𝑋 = (Base‘𝐶) |
3 | | xpcbas.y |
. . . 4
⊢ 𝑌 = (Base‘𝐷) |
4 | | eqid 2736 |
. . . 4
⊢ (Hom
‘𝐶) = (Hom
‘𝐶) |
5 | | eqid 2736 |
. . . 4
⊢ (Hom
‘𝐷) = (Hom
‘𝐷) |
6 | | eqid 2736 |
. . . 4
⊢
(comp‘𝐶) =
(comp‘𝐶) |
7 | | eqid 2736 |
. . . 4
⊢
(comp‘𝐷) =
(comp‘𝐷) |
8 | | simpl 486 |
. . . 4
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐶 ∈ V) |
9 | | simpr 488 |
. . . 4
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝐷 ∈ V) |
10 | | eqidd 2737 |
. . . 4
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (𝑋 × 𝑌)) |
11 | | eqidd 2737 |
. . . 4
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣)))) = (𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣))))) |
12 | | eqidd 2737 |
. . . 4
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑥 ∈ ((𝑋 × 𝑌) × (𝑋 × 𝑌)), 𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑔 ∈ ((2nd ‘𝑥)(𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣))))‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉(comp‘𝐶)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉(comp‘𝐷)(2nd ‘𝑦))(2nd ‘𝑓))〉)) = (𝑥 ∈ ((𝑋 × 𝑌) × (𝑋 × 𝑌)), 𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑔 ∈ ((2nd ‘𝑥)(𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣))))‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉(comp‘𝐶)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉(comp‘𝐷)(2nd ‘𝑦))(2nd ‘𝑓))〉))) |
13 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12 | xpcval 17638 |
. . 3
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = {〈(Base‘ndx),
(𝑋 × 𝑌)〉, 〈(Hom ‘ndx),
(𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣))))〉, 〈(comp‘ndx), (𝑥 ∈ ((𝑋 × 𝑌) × (𝑋 × 𝑌)), 𝑦 ∈ (𝑋 × 𝑌) ↦ (𝑔 ∈ ((2nd ‘𝑥)(𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣))))𝑦), 𝑓 ∈ ((𝑢 ∈ (𝑋 × 𝑌), 𝑣 ∈ (𝑋 × 𝑌) ↦ (((1st ‘𝑢)(Hom ‘𝐶)(1st ‘𝑣)) × ((2nd ‘𝑢)(Hom ‘𝐷)(2nd ‘𝑣))))‘𝑥) ↦ 〈((1st ‘𝑔)(〈(1st
‘(1st ‘𝑥)), (1st ‘(2nd
‘𝑥))〉(comp‘𝐶)(1st ‘𝑦))(1st ‘𝑓)), ((2nd ‘𝑔)(〈(2nd
‘(1st ‘𝑥)), (2nd ‘(2nd
‘𝑥))〉(comp‘𝐷)(2nd ‘𝑦))(2nd ‘𝑓))〉))〉}) |
14 | 2 | fvexi 6709 |
. . . . 5
⊢ 𝑋 ∈ V |
15 | 3 | fvexi 6709 |
. . . . 5
⊢ 𝑌 ∈ V |
16 | 14, 15 | xpex 7516 |
. . . 4
⊢ (𝑋 × 𝑌) ∈ V |
17 | 16 | a1i 11 |
. . 3
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) ∈ V) |
18 | 13, 17 | estrreslem1 17598 |
. 2
⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (Base‘𝑇)) |
19 | | base0 16726 |
. . 3
⊢ ∅ =
(Base‘∅) |
20 | | fvprc 6687 |
. . . . . 6
⊢ (¬
𝐶 ∈ V →
(Base‘𝐶) =
∅) |
21 | 2, 20 | syl5eq 2783 |
. . . . 5
⊢ (¬
𝐶 ∈ V → 𝑋 = ∅) |
22 | | fvprc 6687 |
. . . . . 6
⊢ (¬
𝐷 ∈ V →
(Base‘𝐷) =
∅) |
23 | 3, 22 | syl5eq 2783 |
. . . . 5
⊢ (¬
𝐷 ∈ V → 𝑌 = ∅) |
24 | 21, 23 | orim12i 909 |
. . . 4
⊢ ((¬
𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) → (𝑋 = ∅ ∨ 𝑌 = ∅)) |
25 | | ianor 982 |
. . . 4
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V)) |
26 | | xpeq0 6003 |
. . . 4
⊢ ((𝑋 × 𝑌) = ∅ ↔ (𝑋 = ∅ ∨ 𝑌 = ∅)) |
27 | 24, 25, 26 | 3imtr4i 295 |
. . 3
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = ∅) |
28 | | fnxpc 17637 |
. . . . . . 7
⊢
×c Fn (V × V) |
29 | | fndm 6459 |
. . . . . . 7
⊢ (
×c Fn (V × V) → dom
×c = (V × V)) |
30 | 28, 29 | ax-mp 5 |
. . . . . 6
⊢ dom
×c = (V × V) |
31 | 30 | ndmov 7370 |
. . . . 5
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝐶 ×c
𝐷) =
∅) |
32 | 1, 31 | syl5eq 2783 |
. . . 4
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) → 𝑇 = ∅) |
33 | 32 | fveq2d 6699 |
. . 3
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) →
(Base‘𝑇) =
(Base‘∅)) |
34 | 19, 27, 33 | 3eqtr4a 2797 |
. 2
⊢ (¬
(𝐶 ∈ V ∧ 𝐷 ∈ V) → (𝑋 × 𝑌) = (Base‘𝑇)) |
35 | 18, 34 | pm2.61i 185 |
1
⊢ (𝑋 × 𝑌) = (Base‘𝑇) |