Step | Hyp | Ref
| Expression |
1 | | mbfmco.1 |
. . . . . . 7
⊢ (𝜑 → 𝑅 ∈ ∪ ran
sigAlgebra) |
2 | | mbfmco.2 |
. . . . . . 7
⊢ (𝜑 → 𝑆 ∈ ∪ ran
sigAlgebra) |
3 | | mbfmco2.4 |
. . . . . . 7
⊢ (𝜑 → 𝐹 ∈ (𝑅MblFnM𝑆)) |
4 | 1, 2, 3 | mbfmf 31795 |
. . . . . 6
⊢ (𝜑 → 𝐹:∪ 𝑅⟶∪ 𝑆) |
5 | 4 | ffvelrnda 6864 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → (𝐹‘𝑥) ∈ ∪ 𝑆) |
6 | | mbfmco.3 |
. . . . . . 7
⊢ (𝜑 → 𝑇 ∈ ∪ ran
sigAlgebra) |
7 | | mbfmco2.5 |
. . . . . . 7
⊢ (𝜑 → 𝐺 ∈ (𝑅MblFnM𝑇)) |
8 | 1, 6, 7 | mbfmf 31795 |
. . . . . 6
⊢ (𝜑 → 𝐺:∪ 𝑅⟶∪ 𝑇) |
9 | 8 | ffvelrnda 6864 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → (𝐺‘𝑥) ∈ ∪ 𝑇) |
10 | | opelxpi 5563 |
. . . . 5
⊢ (((𝐹‘𝑥) ∈ ∪ 𝑆 ∧ (𝐺‘𝑥) ∈ ∪ 𝑇) → 〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ (∪
𝑆 × ∪ 𝑇)) |
11 | 5, 9, 10 | syl2anc 587 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → 〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ (∪
𝑆 × ∪ 𝑇)) |
12 | | sxuni 31734 |
. . . . . 6
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran
sigAlgebra) → (∪ 𝑆 × ∪ 𝑇) = ∪
(𝑆 ×s
𝑇)) |
13 | 2, 6, 12 | syl2anc 587 |
. . . . 5
⊢ (𝜑 → (∪ 𝑆
× ∪ 𝑇) = ∪ (𝑆 ×s 𝑇)) |
14 | 13 | adantr 484 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → (∪ 𝑆
× ∪ 𝑇) = ∪ (𝑆 ×s 𝑇)) |
15 | 11, 14 | eleqtrd 2836 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ∪ 𝑅) → 〈(𝐹‘𝑥), (𝐺‘𝑥)〉 ∈ ∪
(𝑆 ×s
𝑇)) |
16 | | mbfmco2.6 |
. . 3
⊢ 𝐻 = (𝑥 ∈ ∪ 𝑅 ↦ 〈(𝐹‘𝑥), (𝐺‘𝑥)〉) |
17 | 15, 16 | fmptd 6891 |
. 2
⊢ (𝜑 → 𝐻:∪ 𝑅⟶∪ (𝑆
×s 𝑇)) |
18 | | eqid 2739 |
. . . . 5
⊢ (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) = (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) |
19 | | vex 3403 |
. . . . . 6
⊢ 𝑎 ∈ V |
20 | | vex 3403 |
. . . . . 6
⊢ 𝑏 ∈ V |
21 | 19, 20 | xpex 7497 |
. . . . 5
⊢ (𝑎 × 𝑏) ∈ V |
22 | 18, 21 | elrnmpo 7305 |
. . . 4
⊢ (𝑐 ∈ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) ↔ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑇 𝑐 = (𝑎 × 𝑏)) |
23 | | simp3 1139 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑐 = (𝑎 × 𝑏)) |
24 | 23 | imaeq2d 5904 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ 𝑐) = (◡𝐻 “ (𝑎 × 𝑏))) |
25 | | simp1 1137 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝜑) |
26 | | simp2l 1200 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑎 ∈ 𝑆) |
27 | | simp2r 1201 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → 𝑏 ∈ 𝑇) |
28 | 4, 8, 16 | xppreima2 30565 |
. . . . . . . . . . 11
⊢ (𝜑 → (◡𝐻 “ (𝑎 × 𝑏)) = ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏))) |
29 | 28 | 3ad2ant1 1134 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐻 “ (𝑎 × 𝑏)) = ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏))) |
30 | 1 | 3ad2ant1 1134 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑅 ∈ ∪ ran
sigAlgebra) |
31 | 2 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑆 ∈ ∪ ran
sigAlgebra) |
32 | 3 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝐹 ∈ (𝑅MblFnM𝑆)) |
33 | | simp2 1138 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑎 ∈ 𝑆) |
34 | 30, 31, 32, 33 | mbfmcnvima 31797 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐹 “ 𝑎) ∈ 𝑅) |
35 | 6 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑇 ∈ ∪ ran
sigAlgebra) |
36 | 7 | 3ad2ant1 1134 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝐺 ∈ (𝑅MblFnM𝑇)) |
37 | | simp3 1139 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → 𝑏 ∈ 𝑇) |
38 | 30, 35, 36, 37 | mbfmcnvima 31797 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐺 “ 𝑏) ∈ 𝑅) |
39 | | inelsiga 31676 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ ∪ ran sigAlgebra ∧ (◡𝐹 “ 𝑎) ∈ 𝑅 ∧ (◡𝐺 “ 𝑏) ∈ 𝑅) → ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)) ∈ 𝑅) |
40 | 30, 34, 38, 39 | syl3anc 1372 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → ((◡𝐹 “ 𝑎) ∩ (◡𝐺 “ 𝑏)) ∈ 𝑅) |
41 | 29, 40 | eqeltrd 2834 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) → (◡𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅) |
42 | 25, 26, 27, 41 | syl3anc 1372 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ (𝑎 × 𝑏)) ∈ 𝑅) |
43 | 24, 42 | eqeltrd 2834 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇) ∧ 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ 𝑐) ∈ 𝑅) |
44 | 43 | 3expia 1122 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑎 ∈ 𝑆 ∧ 𝑏 ∈ 𝑇)) → (𝑐 = (𝑎 × 𝑏) → (◡𝐻 “ 𝑐) ∈ 𝑅)) |
45 | 44 | rexlimdvva 3205 |
. . . . 5
⊢ (𝜑 → (∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑇 𝑐 = (𝑎 × 𝑏) → (◡𝐻 “ 𝑐) ∈ 𝑅)) |
46 | 45 | imp 410 |
. . . 4
⊢ ((𝜑 ∧ ∃𝑎 ∈ 𝑆 ∃𝑏 ∈ 𝑇 𝑐 = (𝑎 × 𝑏)) → (◡𝐻 “ 𝑐) ∈ 𝑅) |
47 | 22, 46 | sylan2b 597 |
. . 3
⊢ ((𝜑 ∧ 𝑐 ∈ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))) → (◡𝐻 “ 𝑐) ∈ 𝑅) |
48 | 47 | ralrimiva 3097 |
. 2
⊢ (𝜑 → ∀𝑐 ∈ ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))(◡𝐻 “ 𝑐) ∈ 𝑅) |
49 | | eqid 2739 |
. . . . 5
⊢ ran
(𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) = ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) |
50 | 49 | txbasex 22320 |
. . . 4
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran
sigAlgebra) → ran (𝑎
∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) ∈ V) |
51 | 2, 6, 50 | syl2anc 587 |
. . 3
⊢ (𝜑 → ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)) ∈ V) |
52 | 49 | sxval 31731 |
. . . 4
⊢ ((𝑆 ∈ ∪ ran sigAlgebra ∧ 𝑇 ∈ ∪ ran
sigAlgebra) → (𝑆
×s 𝑇) =
(sigaGen‘ran (𝑎
∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)))) |
53 | 2, 6, 52 | syl2anc 587 |
. . 3
⊢ (𝜑 → (𝑆 ×s 𝑇) = (sigaGen‘ran (𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏)))) |
54 | 51, 1, 53 | imambfm 31802 |
. 2
⊢ (𝜑 → (𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇)) ↔ (𝐻:∪ 𝑅⟶∪ (𝑆
×s 𝑇)
∧ ∀𝑐 ∈ ran
(𝑎 ∈ 𝑆, 𝑏 ∈ 𝑇 ↦ (𝑎 × 𝑏))(◡𝐻 “ 𝑐) ∈ 𝑅))) |
55 | 17, 48, 54 | mpbir2and 713 |
1
⊢ (𝜑 → 𝐻 ∈ (𝑅MblFnM(𝑆 ×s 𝑇))) |