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Mirrors > Home > MPE Home > Th. List > om2uzsuci | Structured version Visualization version GIF version |
Description: The value of 𝐺 (see om2uz0i 13309) 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 6250 | . . . 4 ⊢ (𝑧 = 𝐴 → suc 𝑧 = suc 𝐴) | |
2 | 1 | fveq2d 6668 | . . 3 ⊢ (𝑧 = 𝐴 → (𝐺‘suc 𝑧) = (𝐺‘suc 𝐴)) |
3 | fveq2 6664 | . . . 4 ⊢ (𝑧 = 𝐴 → (𝐺‘𝑧) = (𝐺‘𝐴)) | |
4 | 3 | oveq1d 7165 | . . 3 ⊢ (𝑧 = 𝐴 → ((𝐺‘𝑧) + 1) = ((𝐺‘𝐴) + 1)) |
5 | 2, 4 | eqeq12d 2837 | . 2 ⊢ (𝑧 = 𝐴 → ((𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1) ↔ (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1))) |
6 | ovex 7183 | . . 3 ⊢ ((𝐺‘𝑧) + 1) ∈ V | |
7 | om2uz.2 | . . . 4 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) | |
8 | oveq1 7157 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 + 1) = (𝑥 + 1)) | |
9 | oveq1 7157 | . . . 4 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 + 1) = ((𝐺‘𝑧) + 1)) | |
10 | 7, 8, 9 | frsucmpt2 8070 | . . 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 3573 | 1 ⊢ (𝐴 ∈ ω → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 Vcvv 3494 ↦ cmpt 5138 ↾ cres 5551 suc csuc 6187 ‘cfv 6349 (class class class)co 7150 ωcom 7574 reccrdg 8039 1c1 10532 + caddc 10534 ℤcz 11975 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 |
This theorem is referenced by: om2uzuzi 13311 om2uzlti 13312 om2uzrani 13314 om2uzrdg 13318 uzrdgsuci 13322 uzrdgxfr 13329 fzennn 13330 axdc4uzlem 13345 hashgadd 13732 |
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