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Mirrors > Home > MPE Home > Th. List > rdglimg | Structured version Visualization version GIF version |
Description: The value of the recursive definition generator at a limit ordinal. (Contributed by NM, 16-Nov-2014.) |
Ref | Expression |
---|---|
rdglimg | ⊢ ((𝐵 ∈ dom rec(𝐹, 𝐴) ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ (rec(𝐹, 𝐴) “ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . 2 ⊢ (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))))) | |
2 | rdgvalg 7861 | . 2 ⊢ (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))) | |
3 | rdgseg 7864 | . 2 ⊢ (𝑦 ∈ dom rec(𝐹, 𝐴) → (rec(𝐹, 𝐴) ↾ 𝑦) ∈ V) | |
4 | rdgfun 7858 | . . 3 ⊢ Fun rec(𝐹, 𝐴) | |
5 | funfn 6220 | . . 3 ⊢ (Fun rec(𝐹, 𝐴) ↔ rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴)) | |
6 | 4, 5 | mpbi 222 | . 2 ⊢ rec(𝐹, 𝐴) Fn dom rec(𝐹, 𝐴) |
7 | rdgdmlim 7859 | . . 3 ⊢ Lim dom rec(𝐹, 𝐴) | |
8 | limord 6090 | . . 3 ⊢ (Lim dom rec(𝐹, 𝐴) → Ord dom rec(𝐹, 𝐴)) | |
9 | 7, 8 | ax-mp 5 | . 2 ⊢ Ord dom rec(𝐹, 𝐴) |
10 | 1, 2, 3, 6, 9 | tz7.44-3 7850 | 1 ⊢ ((𝐵 ∈ dom rec(𝐹, 𝐴) ∧ Lim 𝐵) → (rec(𝐹, 𝐴)‘𝐵) = ∪ (rec(𝐹, 𝐴) “ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1507 ∈ wcel 2050 Vcvv 3415 ∅c0 4180 ifcif 4351 ∪ cuni 4713 ↦ cmpt 5009 dom cdm 5408 ran crn 5409 “ cima 5411 Ord word 6030 Lim wlim 6032 Fun wfun 6184 Fn wfn 6185 ‘cfv 6190 reccrdg 7851 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-reu 3095 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-iun 4795 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-wrecs 7752 df-recs 7814 df-rdg 7852 |
This theorem is referenced by: rdglim 7868 r1limg 8996 |
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