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 8134 |
. . 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 8133 |
. . . . 5
⊢ 𝐺 Fn On |
6 | | fnfun 6479 |
. . . . 5
⊢ (𝐺 Fn On → Fun 𝐺) |
7 | 5, 6 | ax-mp 5 |
. . . 4
⊢ Fun 𝐺 |
8 | | resfunexg 7031 |
. . . 4
⊢ ((Fun
𝐺 ∧ 𝐶 ∈ On) → (𝐺 ↾ 𝐶) ∈ V) |
9 | 7, 1, 8 | sylancr 590 |
. . 3
⊢ (𝜑 → (𝐺 ↾ 𝐶) ∈ V) |
10 | | dmeq 5772 |
. . . . . 6
⊢ (𝑥 = (𝐺 ↾ 𝐶) → dom 𝑥 = dom (𝐺 ↾ 𝐶)) |
11 | 10 | fveq2d 6721 |
. . . . 5
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑅1‘dom
𝑥) =
(𝑅1‘dom (𝐺 ↾ 𝐶))) |
12 | 10 | unieqd 4833 |
. . . . . . 7
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ dom
𝑥 = ∪ dom (𝐺 ↾ 𝐶)) |
13 | 10, 12 | eqeq12d 2753 |
. . . . . 6
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (dom 𝑥 = ∪ dom 𝑥 ↔ dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶))) |
14 | | rneq 5805 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran 𝑥 = ran (𝐺 ↾ 𝐶)) |
15 | | df-ima 5564 |
. . . . . . . . . . . . 13
⊢ (𝐺 “ 𝐶) = ran (𝐺 ↾ 𝐶) |
16 | 14, 15 | eqtr4di 2796 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran 𝑥 = (𝐺 “ 𝐶)) |
17 | 16 | unieqd 4833 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ ran
𝑥 = ∪ (𝐺
“ 𝐶)) |
18 | 17 | rneqd 5807 |
. . . . . . . . . 10
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran ∪
ran 𝑥 = ran ∪ (𝐺
“ 𝐶)) |
19 | 18 | unieqd 4833 |
. . . . . . . . 9
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ ran
∪ ran 𝑥 = ∪ ran ∪ (𝐺
“ 𝐶)) |
20 | | suceq 6278 |
. . . . . . . . 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 7228 |
. . . . . . 7
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (suc ∪
ran ∪ ran 𝑥 ·o (rank‘𝑦)) = (suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦))) |
23 | | fveq1 6716 |
. . . . . . . 8
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑥‘suc (rank‘𝑦)) = ((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))) |
24 | 23 | fveq1d 6719 |
. . . . . . 7
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ((𝑥‘suc (rank‘𝑦))‘𝑦) = (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)) |
25 | 22, 24 | oveq12d 7231 |
. . . . . 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 6724 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑥‘∪ dom 𝑥) = ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) |
28 | 27 | rneqd 5807 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran (𝑥‘∪ dom 𝑥) = ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) |
29 | | oieq2 9129 |
. . . . . . . . . . 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 5744 |
. . . . . . . . 9
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = ◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶)))) |
32 | 31, 27 | coeq12d 5733 |
. . . . . . . 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 5928 |
. . . . . . 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 6721 |
. . . . . 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 4467 |
. . . . 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 5140 |
. . . 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 2737 |
. . . 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 6730 |
. . . . 5
⊢
(𝑅1‘dom (𝐺 ↾ 𝐶)) ∈ V |
39 | 38 | mptex 7039 |
. . . 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 6818 |
. . 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 7568 |
. . . . . . . 8
⊢ (𝐶 ∈ On → 𝐶 ⊆ On) |
43 | 1, 42 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ⊆ On) |
44 | | fnssres 6500 |
. . . . . . 7
⊢ ((𝐺 Fn On ∧ 𝐶 ⊆ On) → (𝐺 ↾ 𝐶) Fn 𝐶) |
45 | 5, 43, 44 | sylancr 590 |
. . . . . 6
⊢ (𝜑 → (𝐺 ↾ 𝐶) Fn 𝐶) |
46 | 45 | fndmd 6483 |
. . . . 5
⊢ (𝜑 → dom (𝐺 ↾ 𝐶) = 𝐶) |
47 | 46 | fveq2d 6721 |
. . . 4
⊢ (𝜑 →
(𝑅1‘dom (𝐺 ↾ 𝐶)) = (𝑅1‘𝐶)) |
48 | 47 | mpteq1d 5144 |
. . 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))))) |
49 | 46 | adantr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → dom (𝐺 ↾ 𝐶) = 𝐶) |
50 | 49 | unieqd 4833 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → ∪ dom (𝐺 ↾ 𝐶) = ∪ 𝐶) |
51 | 49, 50 | eqeq12d 2753 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → (dom (𝐺 ↾ 𝐶) = ∪ dom (𝐺 ↾ 𝐶) ↔ 𝐶 = ∪ 𝐶)) |
52 | 51 | ifbid 4462 |
. . . . 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)))) |
53 | | rankr1ai 9414 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈
(𝑅1‘𝐶) → (rank‘𝑦) ∈ 𝐶) |
54 | 53 | ad2antlr 727 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ 𝐶) |
55 | | simpr 488 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → 𝐶 = ∪ 𝐶) |
56 | 54, 55 | eleqtrd 2840 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ ∪ 𝐶) |
57 | | eloni 6223 |
. . . . . . . . . . . 12
⊢ (𝐶 ∈ On → Ord 𝐶) |
58 | | ordsucuniel 7603 |
. . . . . . . . . . . 12
⊢ (Ord
𝐶 → ((rank‘𝑦) ∈ ∪ 𝐶
↔ suc (rank‘𝑦)
∈ 𝐶)) |
59 | 1, 57, 58 | 3syl 18 |
. . . . . . . . . . 11
⊢ (𝜑 → ((rank‘𝑦) ∈ ∪ 𝐶
↔ suc (rank‘𝑦)
∈ 𝐶)) |
60 | 59 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((rank‘𝑦) ∈ ∪ 𝐶
↔ suc (rank‘𝑦)
∈ 𝐶)) |
61 | 56, 60 | mpbid 235 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → suc (rank‘𝑦) ∈ 𝐶) |
62 | 61 | fvresd 6737 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑦))) |
63 | 62 | fveq1d 6719 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑦)) |
64 | 63 | oveq2d 7229 |
. . . . . 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))) |
65 | 64 | ifeq1da 4470 |
. . . . 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)))) |
66 | 50 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ dom (𝐺 ↾ 𝐶) = ∪ 𝐶) |
67 | 66 | fveq2d 6721 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶)) = ((𝐺 ↾ 𝐶)‘∪ 𝐶)) |
68 | 1 | ad2antrr 726 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 ∈ On) |
69 | | uniexg 7528 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐶 ∈ On → ∪ 𝐶
∈ V) |
70 | | sucidg 6291 |
. . . . . . . . . . . . . . . . 17
⊢ (∪ 𝐶
∈ V → ∪ 𝐶 ∈ suc ∪
𝐶) |
71 | 68, 69, 70 | 3syl 18 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶
∈ suc ∪ 𝐶) |
72 | 1 | adantr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → 𝐶 ∈ On) |
73 | | orduniorsuc 7609 |
. . . . . . . . . . . . . . . . . 18
⊢ (Ord
𝐶 → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶)) |
74 | 72, 57, 73 | 3syl 18 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → (𝐶 = ∪ 𝐶 ∨ 𝐶 = suc ∪ 𝐶)) |
75 | 74 | orcanai 1003 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 = suc ∪ 𝐶) |
76 | 71, 75 | eleqtrrd 2841 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶
∈ 𝐶) |
77 | 76 | fvresd 6737 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ 𝐶) = (𝐺‘∪ 𝐶)) |
78 | 67, 77 | eqtrd 2777 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶)) = (𝐺‘∪ 𝐶)) |
79 | 78 | rneqd 5807 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶)) = ran (𝐺‘∪ 𝐶)) |
80 | | oieq2 9129 |
. . . . . . . . . . . 12
⊢ (ran
((𝐺 ↾ 𝐶)‘∪ dom (𝐺 ↾ 𝐶)) = ran (𝐺‘∪ 𝐶) → OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) = OrdIso( E , ran (𝐺‘∪ 𝐶))) |
81 | 79, 80 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) = OrdIso( E , ran (𝐺‘∪ 𝐶))) |
82 | 81 | cnveqd 5744 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) = ◡OrdIso( E , ran (𝐺‘∪ 𝐶))) |
83 | 82, 78 | coeq12d 5733 |
. . . . . . . . 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))) |
84 | | dfac12.h |
. . . . . . . . 9
⊢ 𝐻 = (◡OrdIso( E , ran (𝐺‘∪ 𝐶)) ∘ (𝐺‘∪ 𝐶)) |
85 | 83, 84 | eqtr4di 2796 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) = 𝐻) |
86 | 85 | imaeq1d 5928 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ((◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) “ 𝑦) = (𝐻 “ 𝑦)) |
87 | 86 | fveq2d 6721 |
. . . . . 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))) |
88 | 87 | ifeq2da 4471 |
. . . . 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)))) |
89 | 52, 65, 88 | 3eqtrd 2781 |
. . . 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)))) |
90 | 89 | mpteq2dva 5150 |
. . 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))))) |
91 | 48, 90 | eqtrd 2777 |
. 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))))) |
92 | 4, 41, 91 | 3eqtrd 2781 |
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))))) |