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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 7796 | . . . . 5 ⊢ Lim dom rec(𝐹, 𝐴) | |
2 | limomss 7348 | . . . . 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 3817 | . . 3 ⊢ (𝐵 ∈ ω → 𝐵 ∈ dom rec(𝐹, 𝐴)) |
5 | rdgsucg 7802 | . . 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 7359 | . . 3 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω) | |
8 | fvres 6465 | . . 3 ⊢ (suc 𝐵 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝐵) = (rec(𝐹, 𝐴)‘suc 𝐵)) | |
9 | 7, 8 | sylbi 209 | . 2 ⊢ (𝐵 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝐵) = (rec(𝐹, 𝐴)‘suc 𝐵)) |
10 | fvres 6465 | . . 3 ⊢ (𝐵 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘𝐵) = (rec(𝐹, 𝐴)‘𝐵)) | |
11 | 10 | fveq2d 6450 | . 2 ⊢ (𝐵 ∈ ω → (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝐵)) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) |
12 | 6, 9, 11 | 3eqtr4d 2824 | 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 1601 ∈ wcel 2107 ⊆ wss 3792 dom cdm 5355 ↾ cres 5357 Lim wlim 5977 suc csuc 5978 ‘cfv 6135 ωcom 7343 reccrdg 7788 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-om 7344 df-wrecs 7689 df-recs 7751 df-rdg 7789 |
This theorem is referenced by: frsucmpt 7816 frsucmptn 7817 seqomlem1 7828 seqomlem4 7831 onasuc 7892 onmsuc 7893 onesuc 7894 inf3lemc 8820 alephfplem2 9261 ackbij2lem2 9397 infpssrlem2 9461 fin23lem34 9503 fin23lem35 9504 itunisuc 9576 om2uzrdg 13074 uzrdgsuci 13078 cnfinltrel 33836 |
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