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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 8357 | . 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) | |
2 | nfcv 2908 | . . . 4 ⊢ Ⅎ𝑥V | |
3 | nfv 1918 | . . . . 5 ⊢ Ⅎ𝑥 𝑔 = ∅ | |
4 | nfrdg.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
5 | nfv 1918 | . . . . . 6 ⊢ Ⅎ𝑥Lim dom 𝑔 | |
6 | nfcv 2908 | . . . . . 6 ⊢ Ⅎ𝑥∪ ran 𝑔 | |
7 | nfrdg.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 | |
8 | nfcv 2908 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑔‘∪ dom 𝑔) | |
9 | 7, 8 | nffv 6853 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘∪ dom 𝑔)) |
10 | 5, 6, 9 | nfif 4517 | . . . . 5 ⊢ Ⅎ𝑥if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))) |
11 | 3, 4, 10 | nfif 4517 | . . . 4 ⊢ Ⅎ𝑥if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))) |
12 | 2, 11 | nfmpt 5213 | . . 3 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) |
13 | 12 | nfrecs 8322 | . 2 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) |
14 | 1, 13 | nfcxfr 2906 | 1 ⊢ Ⅎ𝑥rec(𝐹, 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 Ⅎwnfc 2888 Vcvv 3446 ∅c0 4283 ifcif 4487 ∪ cuni 4866 ↦ cmpt 5189 dom cdm 5634 ran crn 5635 Lim wlim 6319 ‘cfv 6497 recscrecs 8317 reccrdg 8356 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3409 df-v 3448 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-mpt 5190 df-xp 5640 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-iota 6449 df-fv 6505 df-ov 7361 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 |
This theorem is referenced by: rdgsucmptf 8375 rdgsucmptnf 8376 frsucmpt 8385 frsucmptn 8386 ttrclselem1 9662 ttrclselem2 9663 nfseq 13917 rdgssun 35852 exrecfnlem 35853 finxpreclem6 35870 |
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