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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 8055 | . . . . 5 ⊢ Lim dom rec(𝐹, 𝐴) | |
2 | limomss 7587 | . . . . 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 3965 | . . 3 ⊢ (𝐵 ∈ ω → 𝐵 ∈ dom rec(𝐹, 𝐴)) |
5 | rdgsucg 8061 | . . 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 7598 | . . 3 ⊢ (𝐵 ∈ ω ↔ suc 𝐵 ∈ ω) | |
8 | fvres 6691 | . . 3 ⊢ (suc 𝐵 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝐵) = (rec(𝐹, 𝐴)‘suc 𝐵)) | |
9 | 7, 8 | sylbi 219 | . 2 ⊢ (𝐵 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘suc 𝐵) = (rec(𝐹, 𝐴)‘suc 𝐵)) |
10 | fvres 6691 | . . 3 ⊢ (𝐵 ∈ ω → ((rec(𝐹, 𝐴) ↾ ω)‘𝐵) = (rec(𝐹, 𝐴)‘𝐵)) | |
11 | 10 | fveq2d 6676 | . 2 ⊢ (𝐵 ∈ ω → (𝐹‘((rec(𝐹, 𝐴) ↾ ω)‘𝐵)) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) |
12 | 6, 9, 11 | 3eqtr4d 2868 | 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 1537 ∈ wcel 2114 ⊆ wss 3938 dom cdm 5557 ↾ cres 5559 Lim wlim 6194 suc csuc 6195 ‘cfv 6357 ωcom 7582 reccrdg 8047 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-wrecs 7949 df-recs 8010 df-rdg 8048 |
This theorem is referenced by: frsucmpt 8075 frsucmptn 8076 seqomlem1 8088 seqomlem4 8091 onasuc 8155 onmsuc 8156 onesuc 8157 inf3lemc 9091 alephfplem2 9533 ackbij2lem2 9664 infpssrlem2 9728 fin23lem34 9770 fin23lem35 9771 itunisuc 9843 om2uzrdg 13327 uzrdgsuci 13331 |
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