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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 7506 | . 2 ⊢ rec(𝐹, 𝐴) = recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) | |
2 | nfcv 2764 | . . . 4 ⊢ Ⅎ𝑥V | |
3 | nfv 1843 | . . . . 5 ⊢ Ⅎ𝑥 𝑔 = ∅ | |
4 | nfrdg.2 | . . . . 5 ⊢ Ⅎ𝑥𝐴 | |
5 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑥Lim dom 𝑔 | |
6 | nfcv 2764 | . . . . . 6 ⊢ Ⅎ𝑥∪ ran 𝑔 | |
7 | nfrdg.1 | . . . . . . 7 ⊢ Ⅎ𝑥𝐹 | |
8 | nfcv 2764 | . . . . . . 7 ⊢ Ⅎ𝑥(𝑔‘∪ dom 𝑔) | |
9 | 7, 8 | nffv 6198 | . . . . . 6 ⊢ Ⅎ𝑥(𝐹‘(𝑔‘∪ dom 𝑔)) |
10 | 5, 6, 9 | nfif 4115 | . . . . 5 ⊢ Ⅎ𝑥if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))) |
11 | 3, 4, 10 | nfif 4115 | . . . 4 ⊢ Ⅎ𝑥if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))) |
12 | 2, 11 | nfmpt 4746 | . . 3 ⊢ Ⅎ𝑥(𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔))))) |
13 | 12 | nfrecs 7471 | . 2 ⊢ Ⅎ𝑥recs((𝑔 ∈ V ↦ if(𝑔 = ∅, 𝐴, if(Lim dom 𝑔, ∪ ran 𝑔, (𝐹‘(𝑔‘∪ dom 𝑔)))))) |
14 | 1, 13 | nfcxfr 2762 | 1 ⊢ Ⅎ𝑥rec(𝐹, 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 Ⅎwnfc 2751 Vcvv 3200 ∅c0 3915 ifcif 4086 ∪ cuni 4436 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 Lim wlim 5724 ‘cfv 5888 recscrecs 7467 reccrdg 7505 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-xp 5120 df-cnv 5122 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-iota 5851 df-fv 5896 df-wrecs 7407 df-recs 7468 df-rdg 7506 |
This theorem is referenced by: rdgsucmptf 7524 rdgsucmptnf 7525 frsucmpt 7533 frsucmptn 7534 nfseq 12811 trpredlem1 31727 trpredrec 31738 finxpreclem6 33233 |
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