Proof of Theorem domss2
Step | Hyp | Ref
| Expression |
1 | | f1f1orn 6061 |
. . . . . . . 8
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴–1-1-onto→ran
𝐹) |
2 | 1 | 3ad2ant1 1075 |
. . . . . . 7
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹:𝐴–1-1-onto→ran
𝐹) |
3 | | simp2 1055 |
. . . . . . . . . 10
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ∈ 𝑉) |
4 | | rnexg 6990 |
. . . . . . . . . 10
⊢ (𝐴 ∈ 𝑉 → ran 𝐴 ∈ V) |
5 | 3, 4 | syl 17 |
. . . . . . . . 9
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran 𝐴 ∈ V) |
6 | | uniexg 6853 |
. . . . . . . . 9
⊢ (ran
𝐴 ∈ V → ∪ ran 𝐴 ∈ V) |
7 | | pwexg 4776 |
. . . . . . . . 9
⊢ (∪ ran 𝐴 ∈ V → 𝒫 ∪ ran 𝐴 ∈ V) |
8 | 5, 6, 7 | 3syl 18 |
. . . . . . . 8
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝒫 ∪ ran 𝐴 ∈ V) |
9 | | 1stconst 7152 |
. . . . . . . 8
⊢
(𝒫 ∪ ran 𝐴 ∈ V → (1st ↾
((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})–1-1-onto→(𝐵 ∖ ran 𝐹)) |
10 | 8, 9 | syl 17 |
. . . . . . 7
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})–1-1-onto→(𝐵 ∖ ran 𝐹)) |
11 | | difexg 4735 |
. . . . . . . . . 10
⊢ (𝐵 ∈ 𝑊 → (𝐵 ∖ ran 𝐹) ∈ V) |
12 | 11 | 3ad2ant3 1077 |
. . . . . . . . 9
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐵 ∖ ran 𝐹) ∈ V) |
13 | | disjen 8002 |
. . . . . . . . 9
⊢ ((𝐴 ∈ 𝑉 ∧ (𝐵 ∖ ran 𝐹) ∈ V) → ((𝐴 ∩ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) = ∅ ∧ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}) ≈ (𝐵 ∖ ran 𝐹))) |
14 | 3, 12, 13 | syl2anc 691 |
. . . . . . . 8
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐴 ∩ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) = ∅ ∧ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}) ≈ (𝐵 ∖ ran 𝐹))) |
15 | 14 | simpld 474 |
. . . . . . 7
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐴 ∩ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) = ∅) |
16 | | disjdif 3992 |
. . . . . . . 8
⊢ (ran
𝐹 ∩ (𝐵 ∖ ran 𝐹)) = ∅ |
17 | 16 | a1i 11 |
. . . . . . 7
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (ran 𝐹 ∩ (𝐵 ∖ ran 𝐹)) = ∅) |
18 | | f1oun 6069 |
. . . . . . 7
⊢ (((𝐹:𝐴–1-1-onto→ran
𝐹 ∧ (1st
↾ ((𝐵 ∖ ran
𝐹) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})–1-1-onto→(𝐵 ∖ ran 𝐹)) ∧ ((𝐴 ∩ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) = ∅ ∧ (ran 𝐹 ∩ (𝐵 ∖ ran 𝐹)) = ∅)) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))–1-1-onto→(ran
𝐹 ∪ (𝐵 ∖ ran 𝐹))) |
19 | 2, 10, 15, 17, 18 | syl22anc 1319 |
. . . . . 6
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))–1-1-onto→(ran
𝐹 ∪ (𝐵 ∖ ran 𝐹))) |
20 | | undif2 3996 |
. . . . . . . 8
⊢ (ran
𝐹 ∪ (𝐵 ∖ ran 𝐹)) = (ran 𝐹 ∪ 𝐵) |
21 | | f1f 6014 |
. . . . . . . . . . 11
⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) |
22 | 21 | 3ad2ant1 1075 |
. . . . . . . . . 10
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐹:𝐴⟶𝐵) |
23 | | frn 5966 |
. . . . . . . . . 10
⊢ (𝐹:𝐴⟶𝐵 → ran 𝐹 ⊆ 𝐵) |
24 | 22, 23 | syl 17 |
. . . . . . . . 9
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran 𝐹 ⊆ 𝐵) |
25 | | ssequn1 3745 |
. . . . . . . . 9
⊢ (ran
𝐹 ⊆ 𝐵 ↔ (ran 𝐹 ∪ 𝐵) = 𝐵) |
26 | 24, 25 | sylib 207 |
. . . . . . . 8
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (ran 𝐹 ∪ 𝐵) = 𝐵) |
27 | 20, 26 | syl5eq 2656 |
. . . . . . 7
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (ran 𝐹 ∪ (𝐵 ∖ ran 𝐹)) = 𝐵) |
28 | | f1oeq3 6042 |
. . . . . . 7
⊢ ((ran
𝐹 ∪ (𝐵 ∖ ran 𝐹)) = 𝐵 → ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))–1-1-onto→(ran
𝐹 ∪ (𝐵 ∖ ran 𝐹)) ↔ (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))–1-1-onto→𝐵)) |
29 | 27, 28 | syl 17 |
. . . . . 6
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))–1-1-onto→(ran
𝐹 ∪ (𝐵 ∖ ran 𝐹)) ↔ (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))–1-1-onto→𝐵)) |
30 | 19, 29 | mpbid 221 |
. . . . 5
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))–1-1-onto→𝐵) |
31 | | f1ocnv 6062 |
. . . . 5
⊢ ((𝐹 ∪ (1st ↾
((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))–1-1-onto→𝐵 → ◡(𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) |
32 | 30, 31 | syl 17 |
. . . 4
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ◡(𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) |
33 | | domss2.1 |
. . . . 5
⊢ 𝐺 = ◡(𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) |
34 | | f1oeq1 6040 |
. . . . 5
⊢ (𝐺 = ◡(𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) → (𝐺:𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↔ ◡(𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))) |
35 | 33, 34 | ax-mp 5 |
. . . 4
⊢ (𝐺:𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↔ ◡(𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))):𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) |
36 | 32, 35 | sylibr 223 |
. . 3
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐺:𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) |
37 | | f1ofo 6057 |
. . . . 5
⊢ (𝐺:𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) → 𝐺:𝐵–onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) |
38 | | forn 6031 |
. . . . 5
⊢ (𝐺:𝐵–onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) → ran 𝐺 = (𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) |
39 | 36, 37, 38 | 3syl 18 |
. . . 4
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ran 𝐺 = (𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) |
40 | | f1oeq3 6042 |
. . . 4
⊢ (ran
𝐺 = (𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) → (𝐺:𝐵–1-1-onto→ran
𝐺 ↔ 𝐺:𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))) |
41 | 39, 40 | syl 17 |
. . 3
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐺:𝐵–1-1-onto→ran
𝐺 ↔ 𝐺:𝐵–1-1-onto→(𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})))) |
42 | 36, 41 | mpbird 246 |
. 2
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐺:𝐵–1-1-onto→ran
𝐺) |
43 | | ssun1 3738 |
. . 3
⊢ 𝐴 ⊆ (𝐴 ∪ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) |
44 | 43, 39 | syl5sseqr 3617 |
. 2
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → 𝐴 ⊆ ran 𝐺) |
45 | | ssid 3587 |
. . . 4
⊢ ran 𝐹 ⊆ ran 𝐹 |
46 | | cores 5555 |
. . . 4
⊢ (ran
𝐹 ⊆ ran 𝐹 → ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = (𝐺 ∘ 𝐹)) |
47 | 45, 46 | ax-mp 5 |
. . 3
⊢ ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = (𝐺 ∘ 𝐹) |
48 | | dmres 5339 |
. . . . . . . . 9
⊢ dom
(◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) = (ran 𝐹 ∩ dom ◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) |
49 | | f1ocnv 6062 |
. . . . . . . . . . . 12
⊢
((1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})–1-1-onto→(𝐵 ∖ ran 𝐹) → ◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):(𝐵 ∖ ran 𝐹)–1-1-onto→((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) |
50 | | f1odm 6054 |
. . . . . . . . . . . 12
⊢ (◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})):(𝐵 ∖ ran 𝐹)–1-1-onto→((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}) → dom ◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) = (𝐵 ∖ ran 𝐹)) |
51 | 10, 49, 50 | 3syl 18 |
. . . . . . . . . . 11
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → dom ◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) = (𝐵 ∖ ran 𝐹)) |
52 | 51 | ineq2d 3776 |
. . . . . . . . . 10
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (ran 𝐹 ∩ dom ◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) = (ran 𝐹 ∩ (𝐵 ∖ ran 𝐹))) |
53 | 52, 16 | syl6eq 2660 |
. . . . . . . . 9
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (ran 𝐹 ∩ dom ◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) = ∅) |
54 | 48, 53 | syl5eq 2656 |
. . . . . . . 8
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → dom (◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) = ∅) |
55 | | relres 5346 |
. . . . . . . . 9
⊢ Rel
(◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) |
56 | | reldm0 5264 |
. . . . . . . . 9
⊢ (Rel
(◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) → ((◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) = ∅ ↔ dom (◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) = ∅)) |
57 | 55, 56 | ax-mp 5 |
. . . . . . . 8
⊢ ((◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) = ∅ ↔ dom (◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) = ∅) |
58 | 54, 57 | sylibr 223 |
. . . . . . 7
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹) = ∅) |
59 | 58 | uneq2d 3729 |
. . . . . 6
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (◡𝐹 ∪ (◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹)) = (◡𝐹 ∪ ∅)) |
60 | | cnvun 5457 |
. . . . . . . . 9
⊢ ◡(𝐹 ∪ (1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) = (◡𝐹 ∪ ◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) |
61 | 33, 60 | eqtri 2632 |
. . . . . . . 8
⊢ 𝐺 = (◡𝐹 ∪ ◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) |
62 | 61 | reseq1i 5313 |
. . . . . . 7
⊢ (𝐺 ↾ ran 𝐹) = ((◡𝐹 ∪ ◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ↾ ran 𝐹) |
63 | | resundir 5331 |
. . . . . . 7
⊢ ((◡𝐹 ∪ ◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴}))) ↾ ran 𝐹) = ((◡𝐹 ↾ ran 𝐹) ∪ (◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹)) |
64 | | df-rn 5049 |
. . . . . . . . . 10
⊢ ran 𝐹 = dom ◡𝐹 |
65 | 64 | reseq2i 5314 |
. . . . . . . . 9
⊢ (◡𝐹 ↾ ran 𝐹) = (◡𝐹 ↾ dom ◡𝐹) |
66 | | relcnv 5422 |
. . . . . . . . . 10
⊢ Rel ◡𝐹 |
67 | | resdm 5361 |
. . . . . . . . . 10
⊢ (Rel
◡𝐹 → (◡𝐹 ↾ dom ◡𝐹) = ◡𝐹) |
68 | 66, 67 | ax-mp 5 |
. . . . . . . . 9
⊢ (◡𝐹 ↾ dom ◡𝐹) = ◡𝐹 |
69 | 65, 68 | eqtri 2632 |
. . . . . . . 8
⊢ (◡𝐹 ↾ ran 𝐹) = ◡𝐹 |
70 | 69 | uneq1i 3725 |
. . . . . . 7
⊢ ((◡𝐹 ↾ ran 𝐹) ∪ (◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹)) = (◡𝐹 ∪ (◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹)) |
71 | 62, 63, 70 | 3eqtrri 2637 |
. . . . . 6
⊢ (◡𝐹 ∪ (◡(1st ↾ ((𝐵 ∖ ran 𝐹) × {𝒫 ∪ ran 𝐴})) ↾ ran 𝐹)) = (𝐺 ↾ ran 𝐹) |
72 | | un0 3919 |
. . . . . 6
⊢ (◡𝐹 ∪ ∅) = ◡𝐹 |
73 | 59, 71, 72 | 3eqtr3g 2667 |
. . . . 5
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐺 ↾ ran 𝐹) = ◡𝐹) |
74 | 73 | coeq1d 5205 |
. . . 4
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = (◡𝐹 ∘ 𝐹)) |
75 | | f1cocnv1 6079 |
. . . . 5
⊢ (𝐹:𝐴–1-1→𝐵 → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) |
76 | 75 | 3ad2ant1 1075 |
. . . 4
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (◡𝐹 ∘ 𝐹) = ( I ↾ 𝐴)) |
77 | 74, 76 | eqtrd 2644 |
. . 3
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((𝐺 ↾ ran 𝐹) ∘ 𝐹) = ( I ↾ 𝐴)) |
78 | 47, 77 | syl5eqr 2658 |
. 2
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐺 ∘ 𝐹) = ( I ↾ 𝐴)) |
79 | 42, 44, 78 | 3jca 1235 |
1
⊢ ((𝐹:𝐴–1-1→𝐵 ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (𝐺:𝐵–1-1-onto→ran
𝐺 ∧ 𝐴 ⊆ ran 𝐺 ∧ (𝐺 ∘ 𝐹) = ( I ↾ 𝐴))) |