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Mirrors > Home > MPE Home > Th. List > om2uzsuci | Structured version Visualization version GIF version |
Description: The value of 𝐺 (see om2uz0i 13806) at a successor. (Contributed by NM, 3-Oct-2004.) (Revised by Mario Carneiro, 13-Sep-2013.) |
Ref | Expression |
---|---|
om2uz.1 | ⊢ 𝐶 ∈ ℤ |
om2uz.2 | ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) |
Ref | Expression |
---|---|
om2uzsuci | ⊢ (𝐴 ∈ ω → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | suceq 6381 | . . . 4 ⊢ (𝑧 = 𝐴 → suc 𝑧 = suc 𝐴) | |
2 | 1 | fveq2d 6843 | . . 3 ⊢ (𝑧 = 𝐴 → (𝐺‘suc 𝑧) = (𝐺‘suc 𝐴)) |
3 | fveq2 6839 | . . . 4 ⊢ (𝑧 = 𝐴 → (𝐺‘𝑧) = (𝐺‘𝐴)) | |
4 | 3 | oveq1d 7366 | . . 3 ⊢ (𝑧 = 𝐴 → ((𝐺‘𝑧) + 1) = ((𝐺‘𝐴) + 1)) |
5 | 2, 4 | eqeq12d 2752 | . 2 ⊢ (𝑧 = 𝐴 → ((𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1) ↔ (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1))) |
6 | ovex 7384 | . . 3 ⊢ ((𝐺‘𝑧) + 1) ∈ V | |
7 | om2uz.2 | . . . 4 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) | |
8 | oveq1 7358 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 + 1) = (𝑥 + 1)) | |
9 | oveq1 7358 | . . . 4 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 + 1) = ((𝐺‘𝑧) + 1)) | |
10 | 7, 8, 9 | frsucmpt2 8378 | . . 3 ⊢ ((𝑧 ∈ ω ∧ ((𝐺‘𝑧) + 1) ∈ V) → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) |
11 | 6, 10 | mpan2 689 | . 2 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) |
12 | 5, 11 | vtoclga 3532 | 1 ⊢ (𝐴 ∈ ω → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ↦ cmpt 5186 ↾ cres 5633 suc csuc 6317 ‘cfv 6493 (class class class)co 7351 ωcom 7794 reccrdg 8347 1c1 11010 + caddc 11012 ℤcz 12457 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pr 5382 ax-un 7664 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7354 df-om 7795 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 |
This theorem is referenced by: om2uzuzi 13808 om2uzlti 13809 om2uzrani 13811 om2uzrdg 13815 uzrdgsuci 13819 uzrdgxfr 13826 fzennn 13827 axdc4uzlem 13842 hashgadd 14231 |
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