Step | Hyp | Ref
| Expression |
1 | | cnvimass 5485 |
. . . . 5
⊢ (◡𝐹 “ {𝑥}) ⊆ dom 𝐹 |
2 | | dnnumch.f |
. . . . . . 7
⊢ 𝐹 = recs((𝑧 ∈ V ↦ (𝐺‘(𝐴 ∖ ran 𝑧)))) |
3 | 2 | tfr1 7493 |
. . . . . 6
⊢ 𝐹 Fn On |
4 | | fndm 5990 |
. . . . . 6
⊢ (𝐹 Fn On → dom 𝐹 = On) |
5 | 3, 4 | ax-mp 5 |
. . . . 5
⊢ dom 𝐹 = On |
6 | 1, 5 | sseqtri 3637 |
. . . 4
⊢ (◡𝐹 “ {𝑥}) ⊆ On |
7 | | dnnumch.a |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ 𝑉) |
8 | | dnnumch.g |
. . . . . . 7
⊢ (𝜑 → ∀𝑦 ∈ 𝒫 𝐴(𝑦 ≠ ∅ → (𝐺‘𝑦) ∈ 𝑦)) |
9 | 2, 7, 8 | dnnumch2 37615 |
. . . . . 6
⊢ (𝜑 → 𝐴 ⊆ ran 𝐹) |
10 | 9 | sselda 3603 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ran 𝐹) |
11 | | inisegn0 5497 |
. . . . 5
⊢ (𝑥 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑥}) ≠ ∅) |
12 | 10, 11 | sylib 208 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (◡𝐹 “ {𝑥}) ≠ ∅) |
13 | | oninton 7000 |
. . . 4
⊢ (((◡𝐹 “ {𝑥}) ⊆ On ∧ (◡𝐹 “ {𝑥}) ≠ ∅) → ∩ (◡𝐹 “ {𝑥}) ∈ On) |
14 | 6, 12, 13 | sylancr 695 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑥}) ∈ On) |
15 | | eqid 2622 |
. . 3
⊢ (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) = (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})) |
16 | 14, 15 | fmptd 6385 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴⟶On) |
17 | 2, 7, 8 | dnnumch3lem 37616 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣})) |
18 | 17 | adantrr 753 |
. . . . 5
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ∩ (◡𝐹 “ {𝑣})) |
19 | 2, 7, 8 | dnnumch3lem 37616 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤})) |
20 | 19 | adantrl 752 |
. . . . 5
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) = ∩ (◡𝐹 “ {𝑤})) |
21 | 18, 20 | eqeq12d 2637 |
. . . 4
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) ↔ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤}))) |
22 | | fveq2 6191 |
. . . . . . 7
⊢ (∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤}) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = (𝐹‘∩ (◡𝐹 “ {𝑤}))) |
23 | 22 | adantl 482 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = (𝐹‘∩ (◡𝐹 “ {𝑤}))) |
24 | | cnvimass 5485 |
. . . . . . . . . . 11
⊢ (◡𝐹 “ {𝑣}) ⊆ dom 𝐹 |
25 | 24, 5 | sseqtri 3637 |
. . . . . . . . . 10
⊢ (◡𝐹 “ {𝑣}) ⊆ On |
26 | 9 | sselda 3603 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → 𝑣 ∈ ran 𝐹) |
27 | | inisegn0 5497 |
. . . . . . . . . . 11
⊢ (𝑣 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑣}) ≠ ∅) |
28 | 26, 27 | sylib 208 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (◡𝐹 “ {𝑣}) ≠ ∅) |
29 | | onint 6995 |
. . . . . . . . . 10
⊢ (((◡𝐹 “ {𝑣}) ⊆ On ∧ (◡𝐹 “ {𝑣}) ≠ ∅) → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣})) |
30 | 25, 28, 29 | sylancr 695 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣})) |
31 | | fniniseg 6338 |
. . . . . . . . . . 11
⊢ (𝐹 Fn On → (∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}) ↔ (∩
(◡𝐹 “ {𝑣}) ∈ On ∧ (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣))) |
32 | 3, 31 | ax-mp 5 |
. . . . . . . . . 10
⊢ (∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}) ↔ (∩
(◡𝐹 “ {𝑣}) ∈ On ∧ (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣)) |
33 | 32 | simprbi 480 |
. . . . . . . . 9
⊢ (∩ (◡𝐹 “ {𝑣}) ∈ (◡𝐹 “ {𝑣}) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣) |
34 | 30, 33 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐴) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣) |
35 | 34 | adantrr 753 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣) |
36 | 35 | adantr 481 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → (𝐹‘∩ (◡𝐹 “ {𝑣})) = 𝑣) |
37 | | cnvimass 5485 |
. . . . . . . . . . 11
⊢ (◡𝐹 “ {𝑤}) ⊆ dom 𝐹 |
38 | 37, 5 | sseqtri 3637 |
. . . . . . . . . 10
⊢ (◡𝐹 “ {𝑤}) ⊆ On |
39 | 9 | sselda 3603 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → 𝑤 ∈ ran 𝐹) |
40 | | inisegn0 5497 |
. . . . . . . . . . 11
⊢ (𝑤 ∈ ran 𝐹 ↔ (◡𝐹 “ {𝑤}) ≠ ∅) |
41 | 39, 40 | sylib 208 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (◡𝐹 “ {𝑤}) ≠ ∅) |
42 | | onint 6995 |
. . . . . . . . . 10
⊢ (((◡𝐹 “ {𝑤}) ⊆ On ∧ (◡𝐹 “ {𝑤}) ≠ ∅) → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤})) |
43 | 38, 41, 42 | sylancr 695 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → ∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤})) |
44 | | fniniseg 6338 |
. . . . . . . . . . 11
⊢ (𝐹 Fn On → (∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}) ↔ (∩
(◡𝐹 “ {𝑤}) ∈ On ∧ (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤))) |
45 | 3, 44 | ax-mp 5 |
. . . . . . . . . 10
⊢ (∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}) ↔ (∩
(◡𝐹 “ {𝑤}) ∈ On ∧ (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤)) |
46 | 45 | simprbi 480 |
. . . . . . . . 9
⊢ (∩ (◡𝐹 “ {𝑤}) ∈ (◡𝐹 “ {𝑤}) → (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤) |
47 | 43, 46 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑤 ∈ 𝐴) → (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤) |
48 | 47 | adantrl 752 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤) |
49 | 48 | adantr 481 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → (𝐹‘∩ (◡𝐹 “ {𝑤})) = 𝑤) |
50 | 23, 36, 49 | 3eqtr3d 2664 |
. . . . 5
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) ∧ ∩ (◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤})) → 𝑣 = 𝑤) |
51 | 50 | ex 450 |
. . . 4
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (∩
(◡𝐹 “ {𝑣}) = ∩ (◡𝐹 “ {𝑤}) → 𝑣 = 𝑤)) |
52 | 21, 51 | sylbid 230 |
. . 3
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴)) → (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤)) |
53 | 52 | ralrimivva 2971 |
. 2
⊢ (𝜑 → ∀𝑣 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤)) |
54 | | dff13 6512 |
. 2
⊢ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On ↔ ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴⟶On ∧ ∀𝑣 ∈ 𝐴 ∀𝑤 ∈ 𝐴 (((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑣) = ((𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥}))‘𝑤) → 𝑣 = 𝑤))) |
55 | 16, 53, 54 | sylanbrc 698 |
1
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ ∩ (◡𝐹 “ {𝑥})):𝐴–1-1→On) |