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Mirrors > Home > MPE Home > Th. List > nfres | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for restriction. (Contributed by NM, 15-Sep-2003.) (Revised by David Abernethy, 19-Jun-2012.) |
Ref | Expression |
---|---|
nfres.1 | ⊢ Ⅎ𝑥𝐴 |
nfres.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfres | ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-res 5560 | . 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
2 | nfres.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfres.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
4 | nfcv 2974 | . . . 4 ⊢ Ⅎ𝑥V | |
5 | 3, 4 | nfxp 5581 | . . 3 ⊢ Ⅎ𝑥(𝐵 × V) |
6 | 2, 5 | nfin 4190 | . 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V)) |
7 | 1, 6 | nfcxfr 2972 | 1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2958 Vcvv 3492 ∩ cin 3932 × cxp 5546 ↾ cres 5550 |
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-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-in 3940 df-opab 5120 df-xp 5554 df-res 5560 |
This theorem is referenced by: nfima 5930 nfwrecs 7938 frsucmpt 8062 frsucmptn 8063 nfoi 8966 prdsdsf 22904 prdsxmet 22906 limciun 24419 bnj1446 32214 bnj1447 32215 bnj1448 32216 bnj1466 32222 bnj1467 32223 bnj1519 32234 bnj1520 32235 bnj1529 32239 trpredlem1 32963 trpredrec 32974 nffrecs 33017 nosupbnd2 33113 feqresmptf 41375 limcperiod 41785 xlimconst2 41992 cncfiooicclem1 42052 stoweidlem28 42190 nfdfat 43203 setrec2lem2 44725 setrec2 44726 |
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