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Mirrors > Home > MPE Home > Th. List > om2uzsuci | Structured version Visualization version GIF version |
Description: The value of 𝐺 (see om2uz0i 13768) 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 6367 | . . . 4 ⊢ (𝑧 = 𝐴 → suc 𝑧 = suc 𝐴) | |
2 | 1 | fveq2d 6829 | . . 3 ⊢ (𝑧 = 𝐴 → (𝐺‘suc 𝑧) = (𝐺‘suc 𝐴)) |
3 | fveq2 6825 | . . . 4 ⊢ (𝑧 = 𝐴 → (𝐺‘𝑧) = (𝐺‘𝐴)) | |
4 | 3 | oveq1d 7352 | . . 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 7370 | . . 3 ⊢ ((𝐺‘𝑧) + 1) ∈ V | |
7 | om2uz.2 | . . . 4 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) | |
8 | oveq1 7344 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 + 1) = (𝑥 + 1)) | |
9 | oveq1 7344 | . . . 4 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 + 1) = ((𝐺‘𝑧) + 1)) | |
10 | 7, 8, 9 | frsucmpt2 8341 | . . 3 ⊢ ((𝑧 ∈ ω ∧ ((𝐺‘𝑧) + 1) ∈ V) → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) |
11 | 6, 10 | mpan2 688 | . 2 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) |
12 | 5, 11 | vtoclga 3522 | 1 ⊢ (𝐴 ∈ ω → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2105 Vcvv 3441 ↦ cmpt 5175 ↾ cres 5622 suc csuc 6304 ‘cfv 6479 (class class class)co 7337 ωcom 7780 reccrdg 8310 1c1 10973 + caddc 10975 ℤcz 12420 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-sep 5243 ax-nul 5250 ax-pr 5372 ax-un 7650 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-ov 7340 df-om 7781 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 |
This theorem is referenced by: om2uzuzi 13770 om2uzlti 13771 om2uzrani 13773 om2uzrdg 13777 uzrdgsuci 13781 uzrdgxfr 13788 fzennn 13789 axdc4uzlem 13804 hashgadd 14192 |
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