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 8438 |
. . 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 8437 |
. . . . 5
⊢ 𝐺 Fn On |
| 6 | | fnfun 6668 |
. . . . 5
⊢ (𝐺 Fn On → Fun 𝐺) |
| 7 | 5, 6 | ax-mp 5 |
. . . 4
⊢ Fun 𝐺 |
| 8 | | resfunexg 7235 |
. . . 4
⊢ ((Fun
𝐺 ∧ 𝐶 ∈ On) → (𝐺 ↾ 𝐶) ∈ V) |
| 9 | 7, 1, 8 | sylancr 587 |
. . 3
⊢ (𝜑 → (𝐺 ↾ 𝐶) ∈ V) |
| 10 | | dmeq 5914 |
. . . . . 6
⊢ (𝑥 = (𝐺 ↾ 𝐶) → dom 𝑥 = dom (𝐺 ↾ 𝐶)) |
| 11 | 10 | fveq2d 6910 |
. . . . 5
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑅1‘dom
𝑥) =
(𝑅1‘dom (𝐺 ↾ 𝐶))) |
| 12 | 10 | unieqd 4920 |
. . . . . . 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 5947 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran 𝑥 = ran (𝐺 ↾ 𝐶)) |
| 15 | | df-ima 5698 |
. . . . . . . . . . . . 13
⊢ (𝐺 “ 𝐶) = ran (𝐺 ↾ 𝐶) |
| 16 | 14, 15 | eqtr4di 2795 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran 𝑥 = (𝐺 “ 𝐶)) |
| 17 | 16 | unieqd 4920 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ ran
𝑥 = ∪ (𝐺
“ 𝐶)) |
| 18 | 17 | rneqd 5949 |
. . . . . . . . . 10
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran ∪
ran 𝑥 = ran ∪ (𝐺
“ 𝐶)) |
| 19 | 18 | unieqd 4920 |
. . . . . . . . 9
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ∪ ran
∪ ran 𝑥 = ∪ ran ∪ (𝐺
“ 𝐶)) |
| 20 | | suceq 6450 |
. . . . . . . . 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 7446 |
. . . . . . 7
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (suc ∪
ran ∪ ran 𝑥 ·o (rank‘𝑦)) = (suc ∪ ran ∪ (𝐺 “ 𝐶) ·o (rank‘𝑦))) |
| 23 | | fveq1 6905 |
. . . . . . . 8
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑥‘suc (rank‘𝑦)) = ((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))) |
| 24 | 23 | fveq1d 6908 |
. . . . . . 7
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ((𝑥‘suc (rank‘𝑦))‘𝑦) = (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦)) |
| 25 | 22, 24 | oveq12d 7449 |
. . . . . 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 6913 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝐺 ↾ 𝐶) → (𝑥‘∪ dom 𝑥) = ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) |
| 28 | 27 | rneqd 5949 |
. . . . . . . . . . 11
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ran (𝑥‘∪ dom 𝑥) = ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) |
| 29 | | oieq2 9553 |
. . . . . . . . . . 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 5886 |
. . . . . . . . 9
⊢ (𝑥 = (𝐺 ↾ 𝐶) → ◡OrdIso( E , ran (𝑥‘∪ dom 𝑥)) = ◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶)))) |
| 32 | 31, 27 | coeq12d 5875 |
. . . . . . . 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 6077 |
. . . . . . 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 6910 |
. . . . . 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 4554 |
. . . . 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 5233 |
. . . 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 6919 |
. . . . 5
⊢
(𝑅1‘dom (𝐺 ↾ 𝐶)) ∈ V |
| 39 | 38 | mptex 7243 |
. . . 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 7016 |
. . 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 7805 |
. . . . . . . 8
⊢ (𝐶 ∈ On → 𝐶 ⊆ On) |
| 43 | 1, 42 | syl 17 |
. . . . . . 7
⊢ (𝜑 → 𝐶 ⊆ On) |
| 44 | | fnssres 6691 |
. . . . . . 7
⊢ ((𝐺 Fn On ∧ 𝐶 ⊆ On) → (𝐺 ↾ 𝐶) Fn 𝐶) |
| 45 | 5, 43, 44 | sylancr 587 |
. . . . . 6
⊢ (𝜑 → (𝐺 ↾ 𝐶) Fn 𝐶) |
| 46 | 45 | fndmd 6673 |
. . . . 5
⊢ (𝜑 → dom (𝐺 ↾ 𝐶) = 𝐶) |
| 47 | 46 | fveq2d 6910 |
. . . 4
⊢ (𝜑 →
(𝑅1‘dom (𝐺 ↾ 𝐶)) = (𝑅1‘𝐶)) |
| 48 | 47 | mpteq1d 5237 |
. . 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 480 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → dom (𝐺 ↾ 𝐶) = 𝐶) |
| 50 | 49 | unieqd 4920 |
. . . . . . 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 4549 |
. . . . 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 9838 |
. . . . . . . . . . . 12
⊢ (𝑦 ∈
(𝑅1‘𝐶) → (rank‘𝑦) ∈ 𝐶) |
| 54 | 53 | ad2antlr 727 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ 𝐶) |
| 55 | | simpr 484 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → 𝐶 = ∪ 𝐶) |
| 56 | 54, 55 | eleqtrd 2843 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (rank‘𝑦) ∈ ∪ 𝐶) |
| 57 | | eloni 6394 |
. . . . . . . . . . . 12
⊢ (𝐶 ∈ On → Ord 𝐶) |
| 58 | | ordsucuniel 7844 |
. . . . . . . . . . . 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 232 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → suc (rank‘𝑦) ∈ 𝐶) |
| 62 | 61 | fvresd 6926 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → ((𝐺 ↾ 𝐶)‘suc (rank‘𝑦)) = (𝐺‘suc (rank‘𝑦))) |
| 63 | 62 | fveq1d 6908 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ 𝐶 = ∪ 𝐶) → (((𝐺 ↾ 𝐶)‘suc (rank‘𝑦))‘𝑦) = ((𝐺‘suc (rank‘𝑦))‘𝑦)) |
| 64 | 63 | oveq2d 7447 |
. . . . . 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 4557 |
. . . . 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 480 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ dom (𝐺 ↾ 𝐶) = ∪ 𝐶) |
| 67 | 66 | fveq2d 6910 |
. . . . . . . . . . . . . 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 7760 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐶 ∈ On → ∪ 𝐶
∈ V) |
| 70 | | sucidg 6465 |
. . . . . . . . . . . . . . . . 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 480 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) → 𝐶 ∈ On) |
| 73 | | orduniorsuc 7850 |
. . . . . . . . . . . . . . . . . 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 1005 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → 𝐶 = suc ∪ 𝐶) |
| 76 | 71, 75 | eleqtrrd 2844 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ∪ 𝐶
∈ 𝐶) |
| 77 | 76 | fvresd 6926 |
. . . . . . . . . . . . . 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 5949 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶)) = ran (𝐺‘∪ 𝐶)) |
| 80 | | oieq2 9553 |
. . . . . . . . . . . 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 5886 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → ◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) = ◡OrdIso( E , ran (𝐺‘∪ 𝐶))) |
| 83 | 82, 78 | coeq12d 5875 |
. . . . . . . . 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 2795 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑦 ∈ (𝑅1‘𝐶)) ∧ ¬ 𝐶 = ∪ 𝐶) → (◡OrdIso( E , ran ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) ∘ ((𝐺 ↾ 𝐶)‘∪ dom
(𝐺 ↾ 𝐶))) = 𝐻) |
| 86 | 85 | imaeq1d 6077 |
. . . . . . 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 6910 |
. . . . . 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 4558 |
. . . . 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 5242 |
. . 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))))) |