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Mirrors > Home > MPE Home > Th. List > rdgsucg | Structured version Visualization version GIF version |
Description: The value of the recursive definition generator at a successor. (Contributed by NM, 16-Nov-2014.) |
Ref | Expression |
---|---|
rdgsucg | ⊢ (𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rdgdmlim 7796 | . . 3 ⊢ Lim dom rec(𝐹, 𝐴) | |
2 | limsuc 7327 | . . 3 ⊢ (Lim dom rec(𝐹, 𝐴) → (𝐵 ∈ dom rec(𝐹, 𝐴) ↔ suc 𝐵 ∈ dom rec(𝐹, 𝐴))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (𝐵 ∈ dom rec(𝐹, 𝐴) ↔ suc 𝐵 ∈ dom rec(𝐹, 𝐴)) |
4 | eqid 2777 | . . 3 ⊢ (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐹‘(𝑥‘∪ dom 𝑥))))) = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐹‘(𝑥‘∪ dom 𝑥))))) | |
5 | rdgvalg 7798 | . . 3 ⊢ (𝑦 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘𝑦) = ((𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐹‘(𝑥‘∪ dom 𝑥)))))‘(rec(𝐹, 𝐴) ↾ 𝑦))) | |
6 | rdgseg 7801 | . . 3 ⊢ (𝑦 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝑦) ∈ V) | |
7 | rdgfun 7795 | . . . 4 ⊢ Fun rec(𝐹, 𝐴) | |
8 | funfn 6165 | . . . 4 ⊢ (Fun rec(𝐹, 𝐴) ↔ rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴)) | |
9 | 7, 8 | mpbi 222 | . . 3 ⊢ rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴) |
10 | limord 6035 | . . . 4 ⊢ (Lim dom rec(𝐹, 𝐴) → Ord dom rec(𝐹, 𝐴)) | |
11 | 1, 10 | ax-mp 5 | . . 3 ⊢ Ord dom rec(𝐹, 𝐴) |
12 | 4, 5, 6, 9, 11 | tz7.44-2 7786 | . 2 ⊢ (suc 𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) |
13 | 3, 12 | sylbi 209 | 1 ⊢ (𝐵 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴)‘suc 𝐵) = (𝐹‘(rec(𝐹, 𝐴)‘𝐵))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∈ wcel 2106 Vcvv 3397 ∅c0 4140 ifcif 4306 ∪ cuni 4671 ↦ cmpt 4965 dom cdm 5355 ran crn 5356 Ord word 5975 Lim wlim 5977 suc csuc 5978 Fun wfun 6129 Fn wfn 6130 ‘cfv 6135 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 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 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 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-ral 3094 df-rex 3095 df-reu 3096 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 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-wrecs 7689 df-recs 7751 df-rdg 7789 |
This theorem is referenced by: rdgsuc 7803 rdgsucmptnf 7808 frsuc 7815 r1sucg 8929 |
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