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| Mirrors > Home > MPE Home > Th. List > om2uzsuci | Structured version Visualization version GIF version | ||
| Description: The value of 𝐺 (see om2uz0i 13988) 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 6450 | . . . 4 ⊢ (𝑧 = 𝐴 → suc 𝑧 = suc 𝐴) | |
| 2 | 1 | fveq2d 6910 | . . 3 ⊢ (𝑧 = 𝐴 → (𝐺‘suc 𝑧) = (𝐺‘suc 𝐴)) | 
| 3 | fveq2 6906 | . . . 4 ⊢ (𝑧 = 𝐴 → (𝐺‘𝑧) = (𝐺‘𝐴)) | |
| 4 | 3 | oveq1d 7446 | . . 3 ⊢ (𝑧 = 𝐴 → ((𝐺‘𝑧) + 1) = ((𝐺‘𝐴) + 1)) | 
| 5 | 2, 4 | eqeq12d 2753 | . 2 ⊢ (𝑧 = 𝐴 → ((𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1) ↔ (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1))) | 
| 6 | ovex 7464 | . . 3 ⊢ ((𝐺‘𝑧) + 1) ∈ V | |
| 7 | om2uz.2 | . . . 4 ⊢ 𝐺 = (rec((𝑥 ∈ V ↦ (𝑥 + 1)), 𝐶) ↾ ω) | |
| 8 | oveq1 7438 | . . . 4 ⊢ (𝑦 = 𝑥 → (𝑦 + 1) = (𝑥 + 1)) | |
| 9 | oveq1 7438 | . . . 4 ⊢ (𝑦 = (𝐺‘𝑧) → (𝑦 + 1) = ((𝐺‘𝑧) + 1)) | |
| 10 | 7, 8, 9 | frsucmpt2 8480 | . . 3 ⊢ ((𝑧 ∈ ω ∧ ((𝐺‘𝑧) + 1) ∈ V) → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) | 
| 11 | 6, 10 | mpan2 691 | . 2 ⊢ (𝑧 ∈ ω → (𝐺‘suc 𝑧) = ((𝐺‘𝑧) + 1)) | 
| 12 | 5, 11 | vtoclga 3577 | 1 ⊢ (𝐴 ∈ ω → (𝐺‘suc 𝐴) = ((𝐺‘𝐴) + 1)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ↦ cmpt 5225 ↾ cres 5687 suc csuc 6386 ‘cfv 6561 (class class class)co 7431 ωcom 7887 reccrdg 8449 1c1 11156 + caddc 11158 ℤcz 12613 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 | 
| This theorem is referenced by: om2uzuzi 13990 om2uzlti 13991 om2uzrani 13993 om2uzrdg 13997 uzrdgsuci 14001 uzrdgxfr 14008 fzennn 14009 axdc4uzlem 14024 hashgadd 14416 | 
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