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Mirrors > Home > MPE Home > Th. List > om2uzsuci | Structured version Visualization version GIF version |
Description: The value of 𝐺 (see om2uz0i 12608) 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 5707 | . . . 4 ⊢ (𝑧 = 𝐴 → suc 𝑧 = suc 𝐴) | |
2 | 1 | fveq2d 6107 | . . 3 ⊢ (𝑧 = 𝐴 → (𝐺‘suc 𝑧) = (𝐺‘suc 𝐴)) |
3 | fveq2 6103 | . . . 4 ⊢ (𝑧 = 𝐴 → (𝐺‘𝑧) = (𝐺‘𝐴)) | |
4 | 3 | oveq1d 6564 | . . 3 ⊢ (𝑧 = 𝐴 → ((𝐺‘𝑧) + 1) = ((𝐺‘𝐴) + 1)) |
5 | 2, 4 | eqeq12d 2625 | . 2 ⊢ (𝑧 = 𝐴 → ((𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1) ↔ (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1))) |
6 | ovex 6577 | . . 3 ⊢ ((𝐺‘𝑧) + 1) ∈ V | |
7 | om2uz.2 | . . . 4 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) | |
8 | oveq1 6556 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 + 1) = (𝑥 + 1)) | |
9 | oveq1 6556 | . . . 4 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 + 1) = ((𝐺‘𝑧) + 1)) | |
10 | 7, 8, 9 | frsucmpt2 7422 | . . 3 ⊢ ((𝑧 ∈ ω ∧ ((𝐺‘𝑧) + 1) ∈ V) → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) |
11 | 6, 10 | mpan2 703 | . 2 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) |
12 | 5, 11 | vtoclga 3245 | 1 ⊢ (𝐴 ∈ ω → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ↦ cmpt 4643 ↾ cres 5040 suc csuc 5642 ‘cfv 5804 (class class class)co 6549 ωcom 6957 reccrdg 7392 1c1 9816 + caddc 9818 ℤcz 11254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 |
This theorem is referenced by: om2uzuzi 12610 om2uzlti 12611 om2uzrani 12613 om2uzrdg 12617 uzrdgsuci 12621 uzrdgxfr 12628 fzennn 12629 axdc4uzlem 12644 hashgadd 13027 |
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