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| Mirrors > Home > MPE Home > Th. List > frsuc | Structured version Visualization version GIF version | ||
| Description: The successor value resulting from finite recursive definition generation. (Contributed by NM, 15-Oct-1996.) (Revised by Mario Carneiro, 16-Nov-2014.) |
| Ref | Expression |
|---|---|
| frsuc | ⊢ (𝐵 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝐵) = (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝐵))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rdgdmlim 8457 | . . . . 5 ⊢ Lim dom rec(𝐹, 𝐴) | |
| 2 | limomss 7892 | . . . . 5 ⊢ (Lim dom rec(𝐹, 𝐴) → ω ⊆ dom rec(𝐹, 𝐴)) | |
| 3 | 1, 2 | ax-mp 5 | . . . 4 ⊢ ω ⊆ dom rec(𝐹, 𝐴) |
| 4 | 3 | sseli 3979 | . . 3 ⊢ (𝐵 ∈ ω → 𝐵 ∈ dom rec(𝐹, 𝐴)) |
| 5 | rdgsucg 8463 | . . 3 ⊢ (𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) | |
| 6 | 4, 5 | syl 17 | . 2 ⊢ (𝐵 ∈ ω → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) |
| 7 | peano2b 7904 | . . 3 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω) | |
| 8 | fvres 6925 | . . 3 ⊢ (suc 𝐵 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝐵) = (rec(𝐹, 𝐴)‘suc 𝐵)) | |
| 9 | 7, 8 | sylbi 217 | . 2 ⊢ (𝐵 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝐵) = (rec(𝐹, 𝐴)‘suc 𝐵)) |
| 10 | fvres 6925 | . . 3 ⊢ (𝐵 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘𝐵) = (rec(𝐹, 𝐴)‘𝐵)) | |
| 11 | 10 | fveq2d 6910 | . 2 ⊢ (𝐵 ∈ ω → (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝐵)) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) |
| 12 | 6, 9, 11 | 3eqtr4d 2787 | 1 ⊢ (𝐵 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝐵) = (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝐵))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 dom cdm 5685 ↾ cres 5687 Lim wlim 6385 suc csuc 6386 ‘cfv 6561 ωcom 7887 reccrdg 8449 |
| 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: frsucmpt 8478 frsucmptn 8479 seqomlem1 8490 seqomlem4 8493 onasuc 8566 onmsuc 8567 onesuc 8568 inf3lemc 9666 alephfplem2 10145 ackbij2lem2 10279 infpssrlem2 10344 fin23lem34 10386 fin23lem35 10387 itunisuc 10459 om2uzrdg 13997 uzrdgsuci 14001 om2noseqrdg 28310 noseqrdgsuc 28314 |
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