| Step | Hyp | Ref
| Expression |
| 1 | | om2uz.1 |
. . . . . 6
⊢ 𝐶 ∈ ℤ |
| 2 | | om2uz.2 |
. . . . . 6
⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) |
| 3 | | uzrdg.1 |
. . . . . 6
⊢ 𝐴 ∈ V |
| 4 | | uzrdg.2 |
. . . . . 6
⊢ 𝑅 = (rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω) |
| 5 | | uzrdg.3 |
. . . . . 6
⊢ 𝑆 = ran 𝑅 |
| 6 | 1, 2, 3, 4, 5 | uzrdgfni 13999 |
. . . . 5
⊢ 𝑆 Fn
(ℤ≥‘𝐶) |
| 7 | | fnfun 6668 |
. . . . 5
⊢ (𝑆 Fn
(ℤ≥‘𝐶) → Fun 𝑆) |
| 8 | 6, 7 | ax-mp 5 |
. . . 4
⊢ Fun 𝑆 |
| 9 | | peano2uz 12943 |
. . . . . 6
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → (𝐵 + 1) ∈
(ℤ≥‘𝐶)) |
| 10 | 1, 2, 3, 4 | uzrdglem 13998 |
. . . . . 6
⊢ ((𝐵 + 1) ∈
(ℤ≥‘𝐶) → 〈(𝐵 + 1), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))〉 ∈ ran 𝑅) |
| 11 | 9, 10 | syl 17 |
. . . . 5
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → 〈(𝐵 + 1), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))〉 ∈ ran 𝑅) |
| 12 | 11, 5 | eleqtrrdi 2852 |
. . . 4
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → 〈(𝐵 + 1), (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))〉 ∈ 𝑆) |
| 13 | | funopfv 6958 |
. . . 4
⊢ (Fun
𝑆 → (〈(𝐵 + 1), (2nd
‘(𝑅‘(◡𝐺‘(𝐵 + 1))))〉 ∈ 𝑆 → (𝑆‘(𝐵 + 1)) = (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1)))))) |
| 14 | 8, 12, 13 | mpsyl 68 |
. . 3
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → (𝑆‘(𝐵 + 1)) = (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1))))) |
| 15 | 1, 2 | om2uzf1oi 13994 |
. . . . . . . 8
⊢ 𝐺:ω–1-1-onto→(ℤ≥‘𝐶) |
| 16 | | f1ocnvdm 7305 |
. . . . . . . 8
⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (◡𝐺‘𝐵) ∈ ω) |
| 17 | 15, 16 | mpan 690 |
. . . . . . 7
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → (◡𝐺‘𝐵) ∈ ω) |
| 18 | | peano2 7912 |
. . . . . . 7
⊢ ((◡𝐺‘𝐵) ∈ ω → suc (◡𝐺‘𝐵) ∈ ω) |
| 19 | 17, 18 | syl 17 |
. . . . . 6
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → suc (◡𝐺‘𝐵) ∈ ω) |
| 20 | 1, 2 | om2uzsuci 13989 |
. . . . . . . 8
⊢ ((◡𝐺‘𝐵) ∈ ω → (𝐺‘suc (◡𝐺‘𝐵)) = ((𝐺‘(◡𝐺‘𝐵)) + 1)) |
| 21 | 17, 20 | syl 17 |
. . . . . . 7
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → (𝐺‘suc (◡𝐺‘𝐵)) = ((𝐺‘(◡𝐺‘𝐵)) + 1)) |
| 22 | | f1ocnvfv2 7297 |
. . . . . . . . 9
⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ 𝐵 ∈ (ℤ≥‘𝐶)) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵) |
| 23 | 15, 22 | mpan 690 |
. . . . . . . 8
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → (𝐺‘(◡𝐺‘𝐵)) = 𝐵) |
| 24 | 23 | oveq1d 7446 |
. . . . . . 7
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → ((𝐺‘(◡𝐺‘𝐵)) + 1) = (𝐵 + 1)) |
| 25 | 21, 24 | eqtrd 2777 |
. . . . . 6
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → (𝐺‘suc (◡𝐺‘𝐵)) = (𝐵 + 1)) |
| 26 | | f1ocnvfv 7298 |
. . . . . . 7
⊢ ((𝐺:ω–1-1-onto→(ℤ≥‘𝐶) ∧ suc (◡𝐺‘𝐵) ∈ ω) → ((𝐺‘suc (◡𝐺‘𝐵)) = (𝐵 + 1) → (◡𝐺‘(𝐵 + 1)) = suc (◡𝐺‘𝐵))) |
| 27 | 15, 26 | mpan 690 |
. . . . . 6
⊢ (suc
(◡𝐺‘𝐵) ∈ ω → ((𝐺‘suc (◡𝐺‘𝐵)) = (𝐵 + 1) → (◡𝐺‘(𝐵 + 1)) = suc (◡𝐺‘𝐵))) |
| 28 | 19, 25, 27 | sylc 65 |
. . . . 5
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → (◡𝐺‘(𝐵 + 1)) = suc (◡𝐺‘𝐵)) |
| 29 | 28 | fveq2d 6910 |
. . . 4
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → (𝑅‘(◡𝐺‘(𝐵 + 1))) = (𝑅‘suc (◡𝐺‘𝐵))) |
| 30 | 29 | fveq2d 6910 |
. . 3
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → (2nd ‘(𝑅‘(◡𝐺‘(𝐵 + 1)))) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵)))) |
| 31 | 14, 30 | eqtrd 2777 |
. 2
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → (𝑆‘(𝐵 + 1)) = (2nd ‘(𝑅‘suc (◡𝐺‘𝐵)))) |
| 32 | | frsuc 8477 |
. . . . . . . 8
⊢ ((◡𝐺‘𝐵) ∈ ω → ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)‘(◡𝐺‘𝐵)))) |
| 33 | 4 | fveq1i 6907 |
. . . . . . . 8
⊢ (𝑅‘suc (◡𝐺‘𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)‘suc (◡𝐺‘𝐵)) |
| 34 | 4 | fveq1i 6907 |
. . . . . . . . 9
⊢ (𝑅‘(◡𝐺‘𝐵)) = ((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)‘(◡𝐺‘𝐵)) |
| 35 | 34 | fveq2i 6909 |
. . . . . . . 8
⊢ ((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)‘(𝑅‘(◡𝐺‘𝐵))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)‘((rec((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉), 〈𝐶, 𝐴〉) ↾ ω)‘(◡𝐺‘𝐵))) |
| 36 | 32, 33, 35 | 3eqtr4g 2802 |
. . . . . . 7
⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘suc (◡𝐺‘𝐵)) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)‘(𝑅‘(◡𝐺‘𝐵)))) |
| 37 | 1, 2, 3, 4 | om2uzrdg 13997 |
. . . . . . . . 9
⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘(◡𝐺‘𝐵)) = 〈(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉) |
| 38 | 37 | fveq2d 6910 |
. . . . . . . 8
⊢ ((◡𝐺‘𝐵) ∈ ω → ((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)‘(𝑅‘(◡𝐺‘𝐵))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)‘〈(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉)) |
| 39 | | df-ov 7434 |
. . . . . . . 8
⊢ ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = ((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)‘〈(𝐺‘(◡𝐺‘𝐵)), (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉) |
| 40 | 38, 39 | eqtr4di 2795 |
. . . . . . 7
⊢ ((◡𝐺‘𝐵) ∈ ω → ((𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)‘(𝑅‘(◡𝐺‘𝐵))) = ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)(2nd ‘(𝑅‘(◡𝐺‘𝐵))))) |
| 41 | 36, 40 | eqtrd 2777 |
. . . . . 6
⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘suc (◡𝐺‘𝐵)) = ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)(2nd ‘(𝑅‘(◡𝐺‘𝐵))))) |
| 42 | | fvex 6919 |
. . . . . . 7
⊢ (𝐺‘(◡𝐺‘𝐵)) ∈ V |
| 43 | | fvex 6919 |
. . . . . . 7
⊢
(2nd ‘(𝑅‘(◡𝐺‘𝐵))) ∈ V |
| 44 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧 + 1) = ((𝐺‘(◡𝐺‘𝐵)) + 1)) |
| 45 | | oveq1 7438 |
. . . . . . . . 9
⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → (𝑧𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤)) |
| 46 | 44, 45 | opeq12d 4881 |
. . . . . . . 8
⊢ (𝑧 = (𝐺‘(◡𝐺‘𝐵)) → 〈(𝑧 + 1), (𝑧𝐹𝑤)〉 = 〈((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤)〉) |
| 47 | | oveq2 7439 |
. . . . . . . . 9
⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))) |
| 48 | 47 | opeq2d 4880 |
. . . . . . . 8
⊢ (𝑤 = (2nd ‘(𝑅‘(◡𝐺‘𝐵))) → 〈((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹𝑤)〉 = 〈((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))〉) |
| 49 | | oveq1 7438 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑧 → (𝑥 + 1) = (𝑧 + 1)) |
| 50 | | oveq1 7438 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑧 → (𝑥𝐹𝑦) = (𝑧𝐹𝑦)) |
| 51 | 49, 50 | opeq12d 4881 |
. . . . . . . . 9
⊢ (𝑥 = 𝑧 → 〈(𝑥 + 1), (𝑥𝐹𝑦)〉 = 〈(𝑧 + 1), (𝑧𝐹𝑦)〉) |
| 52 | | oveq2 7439 |
. . . . . . . . . 10
⊢ (𝑦 = 𝑤 → (𝑧𝐹𝑦) = (𝑧𝐹𝑤)) |
| 53 | 52 | opeq2d 4880 |
. . . . . . . . 9
⊢ (𝑦 = 𝑤 → 〈(𝑧 + 1), (𝑧𝐹𝑦)〉 = 〈(𝑧 + 1), (𝑧𝐹𝑤)〉) |
| 54 | 51, 53 | cbvmpov 7528 |
. . . . . . . 8
⊢ (𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉) = (𝑧 ∈ V, 𝑤 ∈ V ↦ 〈(𝑧 + 1), (𝑧𝐹𝑤)〉) |
| 55 | | opex 5469 |
. . . . . . . 8
⊢
〈((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))〉 ∈ V |
| 56 | 46, 48, 54, 55 | ovmpo 7593 |
. . . . . . 7
⊢ (((𝐺‘(◡𝐺‘𝐵)) ∈ V ∧ (2nd
‘(𝑅‘(◡𝐺‘𝐵))) ∈ V) → ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = 〈((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))〉) |
| 57 | 42, 43, 56 | mp2an 692 |
. . . . . 6
⊢ ((𝐺‘(◡𝐺‘𝐵))(𝑥 ∈ V, 𝑦 ∈ V ↦ 〈(𝑥 + 1), (𝑥𝐹𝑦)〉)(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = 〈((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))〉 |
| 58 | 41, 57 | eqtrdi 2793 |
. . . . 5
⊢ ((◡𝐺‘𝐵) ∈ ω → (𝑅‘suc (◡𝐺‘𝐵)) = 〈((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))〉) |
| 59 | 58 | fveq2d 6910 |
. . . 4
⊢ ((◡𝐺‘𝐵) ∈ ω → (2nd
‘(𝑅‘suc (◡𝐺‘𝐵))) = (2nd ‘〈((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))〉)) |
| 60 | | ovex 7464 |
. . . . 5
⊢ ((𝐺‘(◡𝐺‘𝐵)) + 1) ∈ V |
| 61 | | ovex 7464 |
. . . . 5
⊢ ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) ∈ V |
| 62 | 60, 61 | op2nd 8023 |
. . . 4
⊢
(2nd ‘〈((𝐺‘(◡𝐺‘𝐵)) + 1), ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))〉) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) |
| 63 | 59, 62 | eqtrdi 2793 |
. . 3
⊢ ((◡𝐺‘𝐵) ∈ ω → (2nd
‘(𝑅‘suc (◡𝐺‘𝐵))) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))) |
| 64 | 17, 63 | syl 17 |
. 2
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → (2nd ‘(𝑅‘suc (◡𝐺‘𝐵))) = ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵))))) |
| 65 | 1, 2, 3, 4 | uzrdglem 13998 |
. . . . . 6
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ ran 𝑅) |
| 66 | 65, 5 | eleqtrrdi 2852 |
. . . . 5
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → 〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ 𝑆) |
| 67 | | funopfv 6958 |
. . . . 5
⊢ (Fun
𝑆 → (〈𝐵, (2nd ‘(𝑅‘(◡𝐺‘𝐵)))〉 ∈ 𝑆 → (𝑆‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵))))) |
| 68 | 8, 66, 67 | mpsyl 68 |
. . . 4
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → (𝑆‘𝐵) = (2nd ‘(𝑅‘(◡𝐺‘𝐵)))) |
| 69 | 68 | eqcomd 2743 |
. . 3
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → (2nd ‘(𝑅‘(◡𝐺‘𝐵))) = (𝑆‘𝐵)) |
| 70 | 23, 69 | oveq12d 7449 |
. 2
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → ((𝐺‘(◡𝐺‘𝐵))𝐹(2nd ‘(𝑅‘(◡𝐺‘𝐵)))) = (𝐵𝐹(𝑆‘𝐵))) |
| 71 | 31, 64, 70 | 3eqtrd 2781 |
1
⊢ (𝐵 ∈
(ℤ≥‘𝐶) → (𝑆‘(𝐵 + 1)) = (𝐵𝐹(𝑆‘𝐵))) |