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Mirrors > Home > MPE Home > Th. List > nfrdg | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for the recursive definition generator. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 8-Sep-2013.) |
Ref | Expression |
---|---|
nfrdg.1 | ⊢ Ⅎ𝑥𝐹 |
nfrdg.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfrdg | ⊢ Ⅎ𝑥rec(𝐹, 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rdg 8035 | . 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) | |
2 | nfcv 2974 | . . . 4 ⊢ Ⅎ𝑥V | |
3 | nfv 1906 | . . . . 5 ⊢ Ⅎ𝑥 𝑔 = ∅ | |
4 | nfrdg.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
5 | nfv 1906 | . . . . . 6 ⊢ Ⅎ𝑥Lim dom 𝑔 | |
6 | nfcv 2974 | . . . . . 6 ⊢ Ⅎ𝑥∪ ran 𝑔 | |
7 | nfrdg.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 | |
8 | nfcv 2974 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑔‘∪ dom 𝑔) | |
9 | 7, 8 | nffv 6673 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘∪ dom 𝑔)) |
10 | 5, 6, 9 | nfif 4492 | . . . . 5 ⊢ Ⅎ𝑥if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))) |
11 | 3, 4, 10 | nfif 4492 | . . . 4 ⊢ Ⅎ𝑥if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))) |
12 | 2, 11 | nfmpt 5154 | . . 3 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) |
13 | 12 | nfrecs 8000 | . 2 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) |
14 | 1, 13 | nfcxfr 2972 | 1 ⊢ Ⅎ𝑥rec(𝐹, 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 Ⅎwnfc 2958 Vcvv 3492 ∅c0 4288 ifcif 4463 ∪ cuni 4830 ↦ cmpt 5137 dom cdm 5548 ran crn 5549 Lim wlim 6185 ‘cfv 6348 recscrecs 7996 reccrdg 8034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-xp 5554 df-cnv 5556 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-iota 6307 df-fv 6356 df-wrecs 7936 df-recs 7997 df-rdg 8035 |
This theorem is referenced by: rdgsucmptf 8053 rdgsucmptnf 8054 frsucmpt 8062 frsucmptn 8063 nfseq 13367 trpredlem1 32963 trpredrec 32974 rdgssun 34541 exrecfnlem 34542 finxpreclem6 34559 |
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