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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 8241 | . 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) | |
2 | nfcv 2907 | . . . 4 ⊢ Ⅎ𝑥V | |
3 | nfv 1917 | . . . . 5 ⊢ Ⅎ𝑥 𝑔 = ∅ | |
4 | nfrdg.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
5 | nfv 1917 | . . . . . 6 ⊢ Ⅎ𝑥Lim dom 𝑔 | |
6 | nfcv 2907 | . . . . . 6 ⊢ Ⅎ𝑥∪ ran 𝑔 | |
7 | nfrdg.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 | |
8 | nfcv 2907 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑔‘∪ dom 𝑔) | |
9 | 7, 8 | nffv 6784 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘∪ dom 𝑔)) |
10 | 5, 6, 9 | nfif 4489 | . . . . 5 ⊢ Ⅎ𝑥if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))) |
11 | 3, 4, 10 | nfif 4489 | . . . 4 ⊢ Ⅎ𝑥if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))) |
12 | 2, 11 | nfmpt 5181 | . . 3 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) |
13 | 12 | nfrecs 8206 | . 2 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) |
14 | 1, 13 | nfcxfr 2905 | 1 ⊢ Ⅎ𝑥rec(𝐹, 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 Ⅎwnfc 2887 Vcvv 3432 ∅c0 4256 ifcif 4459 ∪ cuni 4839 ↦ cmpt 5157 dom cdm 5589 ran crn 5590 Lim wlim 6267 ‘cfv 6433 recscrecs 8201 reccrdg 8240 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-xp 5595 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-iota 6391 df-fv 6441 df-ov 7278 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 |
This theorem is referenced by: rdgsucmptf 8259 rdgsucmptnf 8260 frsucmpt 8269 frsucmptn 8270 ttrclselem1 9483 ttrclselem2 9484 nfseq 13731 rdgssun 35549 exrecfnlem 35550 finxpreclem6 35567 |
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