Proof of Theorem dfac12lem1
Step | Hyp | Ref
| Expression |
1 | | dfac12.5 |
. . 3
⊢ (𝜑 → 𝐶 ∈ On) |
2 | | dfac12.4 |
. . . 4
⊢ 𝐺 = recs((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom
𝑥) ↦ if(dom 𝑥 = ∪
dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·𝑜
(rank‘𝑦))
+𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦)))))) |
3 | 2 | tfr2 7381 |
. . 3
⊢ (𝐶 ∈ On → (𝐺‘𝐶) = ((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom
𝑥) ↦ if(dom 𝑥 = ∪
dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·𝑜
(rank‘𝑦))
+𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦)))))‘(𝐺 ↾ 𝐶))) |
4 | 1, 3 | syl 17 |
. 2
⊢ (𝜑 → (𝐺‘𝐶) = ((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom
𝑥) ↦ if(dom 𝑥 = ∪
dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·𝑜
(rank‘𝑦))
+𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦)))))‘(𝐺 ↾ 𝐶))) |
5 | 2 | tfr1 7380 |
. . . . 5
⊢ 𝐺 Fn On |
6 | | fnfun 5902 |
. . . . 5
⊢ (𝐺 Fn On → Fun 𝐺) |
7 | 5, 6 | ax-mp 5 |
. . . 4
⊢ Fun 𝐺 |
8 | | resfunexg 6384 |
. . . 4
⊢ ((Fun
𝐺 ∧ 𝐶 ∈ On) → (𝐺 ↾ 𝐶) ∈ V) |
9 | 7, 1, 8 | sylancr 694 |
. . 3
⊢ (𝜑 → (𝐺 ↾ 𝐶) ∈ V) |
10 | | dmeq 5246 |
. . . . . 6
⊢ (𝑥 = (𝐺 ↾ 𝐶) → dom 𝑥 = dom (𝐺 ↾ 𝐶)) |
11 | 10 | fveq2d 6107 |
. . . . 5
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑅1‘dom
𝑥) =
(𝑅1‘dom (𝐺 ↾ 𝐶))) |
12 | 10 | unieqd 4382 |
. . . . . . 7
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ dom
𝑥 = ∪ dom (𝐺 ↾ 𝐶)) |
13 | 10, 12 | eqeq12d 2625 |
. . . . . 6
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (dom 𝑥 = ∪ dom 𝑥 ↔ dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶))) |
14 | | rneq 5272 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran 𝑥 = ran (𝐺 ↾ 𝐶)) |
15 | | df-ima 5051 |
. . . . . . . . . . . . 13
⊢ (𝐺 “ 𝐶) = ran (𝐺 ↾ 𝐶) |
16 | 14, 15 | syl6eqr 2662 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran 𝑥 = (𝐺 “ 𝐶)) |
17 | 16 | unieqd 4382 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ ran
𝑥 = ∪ (𝐺
“ 𝐶)) |
18 | 17 | rneqd 5274 |
. . . . . . . . . 10
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran ∪
ran 𝑥 = ran ∪ (𝐺
“ 𝐶)) |
19 | 18 | unieqd 4382 |
. . . . . . . . 9
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ ran
∪ ran 𝑥 = ∪ ran ∪ (𝐺
“ 𝐶)) |
20 | | suceq 5707 |
. . . . . . . . 9
⊢ (∪ ran ∪ ran 𝑥 = ∪ ran ∪ (𝐺
“ 𝐶) → suc ∪ ran ∪ ran 𝑥 = suc ∪ ran ∪ (𝐺
“ 𝐶)) |
21 | 19, 20 | syl 17 |
. . . . . . . 8
⊢ (𝑥 = (𝐺 ↾ 𝐶) → suc ∪
ran ∪ ran 𝑥 = suc ∪ ran ∪ (𝐺
“ 𝐶)) |
22 | 21 | oveq1d 6564 |
. . . . . . 7
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (suc ∪
ran ∪ ran 𝑥 ·𝑜
(rank‘𝑦)) = (suc
∪ ran ∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))) |
23 | | fveq1 6102 |
. . . . . . . 8
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑥‘suc (rank‘𝑦)) = ((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))) |
24 | 23 | fveq1d 6105 |
. . . . . . 7
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ((𝑥‘suc (rank‘𝑦))‘𝑦) = (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)) |
25 | 22, 24 | oveq12d 6567 |
. . . . . 6
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ((suc ∪
ran ∪ ran 𝑥 ·𝑜
(rank‘𝑦))
+𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)) = ((suc ∪ ran
∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 (((𝐺
↾ 𝐶)‘suc
(rank‘𝑦))‘𝑦))) |
26 | | id 22 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝐺 ↾ 𝐶) → 𝑥 = (𝐺 ↾ 𝐶)) |
27 | 26, 12 | fveq12d 6109 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑥‘∪ dom 𝑥) = ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) |
28 | 27 | rneqd 5274 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran (𝑥‘∪ dom 𝑥) = ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) |
29 | | oieq2 8301 |
. . . . . . . . . . 11
⊢ (ran
(𝑥‘∪ dom 𝑥) = ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶)) → OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶)))) |
30 | 28, 29 | syl 17 |
. . . . . . . . . 10
⊢ (𝑥 = (𝐺 ↾ 𝐶) → OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶)))) |
31 | 30 | cnveqd 5220 |
. . . . . . . . 9
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = ◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶)))) |
32 | 31, 27 | coeq12d 5208 |
. . . . . . . 8
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) = (◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶)))) |
33 | 32 | imaeq1d 5384 |
. . . . . . 7
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦) = ((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦)) |
34 | 33 | fveq2d 6107 |
. . . . . 6
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦)) = (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦))) |
35 | 13, 25, 34 | ifbieq12d 4063 |
. . . . 5
⊢ (𝑥 = (𝐺 ↾ 𝐶) → if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·𝑜
(rank‘𝑦))
+𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦))) = if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran
∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 (((𝐺
↾ 𝐶)‘suc
(rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦)))) |
36 | 11, 35 | mpteq12dv 4663 |
. . . 4
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑦 ∈ (𝑅1‘dom
𝑥) ↦ if(dom 𝑥 = ∪
dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·𝑜
(rank‘𝑦))
+𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦)))) = (𝑦 ∈ (𝑅1‘dom
(𝐺 ↾ 𝐶)) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran
∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 (((𝐺
↾ 𝐶)‘suc
(rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦))))) |
37 | | eqid 2610 |
. . . 4
⊢ (𝑥 ∈ V ↦ (𝑦 ∈
(𝑅1‘dom 𝑥) ↦ if(dom 𝑥 = ∪ dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·𝑜
(rank‘𝑦))
+𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦))))) = (𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom
𝑥) ↦ if(dom 𝑥 = ∪
dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·𝑜
(rank‘𝑦))
+𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦))))) |
38 | | fvex 6113 |
. . . . 5
⊢
(𝑅1‘dom (𝐺 ↾ 𝐶)) ∈ V |
39 | 38 | mptex 6390 |
. . . 4
⊢ (𝑦 ∈
(𝑅1‘dom (𝐺 ↾ 𝐶)) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran
∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 (((𝐺
↾ 𝐶)‘suc
(rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦)))) ∈ V |
40 | 36, 37, 39 | fvmpt 6191 |
. . 3
⊢ ((𝐺 ↾ 𝐶) ∈ V → ((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom
𝑥) ↦ if(dom 𝑥 = ∪
dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·𝑜
(rank‘𝑦))
+𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦)))))‘(𝐺 ↾ 𝐶)) = (𝑦 ∈ (𝑅1‘dom
(𝐺 ↾ 𝐶)) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran
∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 (((𝐺
↾ 𝐶)‘suc
(rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦))))) |
41 | 9, 40 | syl 17 |
. 2
⊢ (𝜑 → ((𝑥 ∈ V ↦ (𝑦 ∈ (𝑅1‘dom
𝑥) ↦ if(dom 𝑥 = ∪
dom 𝑥, ((suc ∪ ran ∪ ran 𝑥 ·𝑜
(rank‘𝑦))
+𝑜 ((𝑥‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) ∘ (𝑥‘∪ dom 𝑥)) “ 𝑦)))))‘(𝐺 ↾ 𝐶)) = (𝑦 ∈ (𝑅1‘dom
(𝐺 ↾ 𝐶)) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran
∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 (((𝐺
↾ 𝐶)‘suc
(rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦))))) |
42 | | onss 6882 |
. . . . . . . 8
⊢ (𝐶 ∈ On → 𝐶 ⊆ On) |
43 | 1, 42 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ⊆ On) |
44 | | fnssres 5918 |
. . . . . . 7
⊢ ((𝐺 Fn On ∧ 𝐶 ⊆ On) → (𝐺 ↾ 𝐶) Fn 𝐶) |
45 | 5, 43, 44 | sylancr 694 |
. . . . . 6
⊢ (𝜑 → (𝐺 ↾ 𝐶) Fn 𝐶) |
46 | | fndm 5904 |
. . . . . 6
⊢ ((𝐺 ↾ 𝐶) Fn 𝐶 → dom (𝐺 ↾ 𝐶) = 𝐶) |
47 | 45, 46 | syl 17 |
. . . . 5
⊢ (𝜑 → dom (𝐺 ↾ 𝐶) = 𝐶) |
48 | 47 | fveq2d 6107 |
. . . 4
⊢ (𝜑 →
(𝑅1‘dom (𝐺 ↾ 𝐶)) = (𝑅1‘𝐶)) |
49 | 48 | mpteq1d 4666 |
. . 3
⊢ (𝜑 → (𝑦 ∈ (𝑅1‘dom
(𝐺 ↾ 𝐶)) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran
∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 (((𝐺
↾ 𝐶)‘suc
(rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦)))) = (𝑦 ∈ (𝑅1‘𝐶) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran
∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 (((𝐺
↾ 𝐶)‘suc
(rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦))))) |
50 | 47 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → dom (𝐺 ↾ 𝐶) = 𝐶) |
51 | 50 | unieqd 4382 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → ∪ dom (𝐺 ↾ 𝐶) = ∪ 𝐶) |
52 | 50, 51 | eqeq12d 2625 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → (dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶) ↔ 𝐶 = ∪ 𝐶)) |
53 | 52 | ifbid 4058 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran
∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 (((𝐺
↾ 𝐶)‘suc
(rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 (((𝐺
↾ 𝐶)‘suc
(rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦)))) |
54 | | rankr1ai 8544 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈
(𝑅1‘𝐶) → (rank‘𝑦) ∈ 𝐶) |
55 | 54 | ad2antlr 759 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ 𝐶) |
56 | | simpr 476 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → 𝐶 = ∪ 𝐶) |
57 | 55, 56 | eleqtrd 2690 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ ∪ 𝐶) |
58 | | eloni 5650 |
. . . . . . . . . . . 12
⊢ (𝐶 ∈ On → Ord 𝐶) |
59 | | ordsucuniel 6916 |
. . . . . . . . . . . 12
⊢ (Ord
𝐶 → ((rank‘𝑦) ∈ ∪ 𝐶
↔ suc (rank‘𝑦)
∈ 𝐶)) |
60 | 1, 58, 59 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → ((rank‘𝑦) ∈ ∪ 𝐶
↔ suc (rank‘𝑦)
∈ 𝐶)) |
61 | 60 | ad2antrr 758 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((rank‘𝑦) ∈ ∪ 𝐶
↔ suc (rank‘𝑦)
∈ 𝐶)) |
62 | 57, 61 | mpbid 221 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → suc (rank‘𝑦) ∈ 𝐶) |
63 | | fvres 6117 |
. . . . . . . . 9
⊢ (suc
(rank‘𝑦) ∈ 𝐶 → ((𝐺 ↾ 𝐶)‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑦))) |
64 | 62, 63 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑦))) |
65 | 64 | fveq1d 6105 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑦)) |
66 | 65 | oveq2d 6565 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 (((𝐺
↾ 𝐶)‘suc
(rank‘𝑦))‘𝑦)) = ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦))) |
67 | 66 | ifeq1da 4066 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 (((𝐺
↾ 𝐶)‘suc
(rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦)))) |
68 | 51 | adantr 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ dom (𝐺 ↾ 𝐶) = ∪ 𝐶) |
69 | 68 | fveq2d 6107 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶)) = ((𝐺 ↾ 𝐶)‘∪ 𝐶)) |
70 | 1 | ad2antrr 758 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 ∈ On) |
71 | | uniexg 6853 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐶 ∈ On → ∪ 𝐶
∈ V) |
72 | | sucidg 5720 |
. . . . . . . . . . . . . . . . 17
⊢ (∪ 𝐶
∈ V → ∪ 𝐶 ∈ suc ∪
𝐶) |
73 | 70, 71, 72 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶
∈ suc ∪ 𝐶) |
74 | 1 | adantr 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → 𝐶 ∈ On) |
75 | | orduniorsuc 6922 |
. . . . . . . . . . . . . . . . . 18
⊢ (Ord
𝐶 → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶)) |
76 | 74, 58, 75 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶)) |
77 | 76 | orcanai 950 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 = suc ∪ 𝐶) |
78 | 73, 77 | eleqtrrd 2691 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶
∈ 𝐶) |
79 | | fvres 6117 |
. . . . . . . . . . . . . . 15
⊢ (∪ 𝐶
∈ 𝐶 → ((𝐺 ↾ 𝐶)‘∪ 𝐶) = (𝐺‘∪ 𝐶)) |
80 | 78, 79 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ 𝐶) = (𝐺‘∪ 𝐶)) |
81 | 69, 80 | eqtrd 2644 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶)) = (𝐺‘∪ 𝐶)) |
82 | 81 | rneqd 5274 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶)) = ran (𝐺‘∪ 𝐶)) |
83 | | oieq2 8301 |
. . . . . . . . . . . 12
⊢ (ran
((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)) = ran (𝐺‘∪ 𝐶) → OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) = OrdIso( E , ran (𝐺‘∪ 𝐶))) |
84 | 82, 83 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) = OrdIso( E , ran (𝐺‘∪ 𝐶))) |
85 | 84 | cnveqd 5220 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) = ◡OrdIso( E , ran (𝐺‘∪ 𝐶))) |
86 | 85, 81 | coeq12d 5208 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) = (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶))) |
87 | | dfac12.h |
. . . . . . . . 9
⊢ 𝐻 = (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶)) |
88 | 86, 87 | syl6eqr 2662 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) = 𝐻) |
89 | 88 | imaeq1d 5384 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦) = (𝐻 “ 𝑦)) |
90 | 89 | fveq2d 6107 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦)) = (𝐹‘(𝐻 “ 𝑦))) |
91 | 90 | ifeq2da 4067 |
. . . . 5
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))) |
92 | 53, 67, 91 | 3eqtrd 2648 |
. . . 4
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran
∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 (((𝐺
↾ 𝐶)‘suc
(rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦))) = if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦)))) |
93 | 92 | mpteq2dva 4672 |
. . 3
⊢ (𝜑 → (𝑦 ∈ (𝑅1‘𝐶) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran
∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 (((𝐺
↾ 𝐶)‘suc
(rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦)))) = (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))))) |
94 | 49, 93 | eqtrd 2644 |
. 2
⊢ (𝜑 → (𝑦 ∈ (𝑅1‘dom
(𝐺 ↾ 𝐶)) ↦ if(dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶), ((suc ∪ ran
∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 (((𝐺
↾ 𝐶)‘suc
(rank‘𝑦))‘𝑦)), (𝐹‘((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦)))) = (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))))) |
95 | 4, 41, 94 | 3eqtrd 2648 |
1
⊢ (𝜑 → (𝐺‘𝐶) = (𝑦 ∈ (𝑅1‘𝐶) ↦ if(𝐶 = ∪ 𝐶, ((suc ∪ ran ∪ (𝐺 “ 𝐶) ·𝑜
(rank‘𝑦))
+𝑜 ((𝐺‘suc (rank‘𝑦))‘𝑦)), (𝐹‘(𝐻 “ 𝑦))))) |