| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| 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 5697 | . 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
| 2 | nfres.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
| 3 | nfres.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
| 4 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥V | |
| 5 | 3, 4 | nfxp 5718 | . . 3 ⊢ Ⅎ𝑥(𝐵 × V) |
| 6 | 2, 5 | nfin 4224 | . 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V)) |
| 7 | 1, 6 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: Ⅎwnfc 2890 Vcvv 3480 ∩ cin 3950 × cxp 5683 ↾ cres 5687 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-v 3482 df-in 3958 df-opab 5206 df-xp 5691 df-res 5697 |
| This theorem is referenced by: nfima 6086 nffrecs 8308 nfwrecsOLD 8342 frsucmpt 8478 frsucmptn 8479 nfoi 9554 prdsdsf 24377 prdsxmet 24379 limciun 25929 nosupbnd2 27761 noinfbnd2 27776 2ndresdju 32659 gsumpart 33060 bnj1446 35059 bnj1447 35060 bnj1448 35061 bnj1466 35067 bnj1467 35068 bnj1519 35079 bnj1520 35080 bnj1529 35084 feqresmptf 45236 limcperiod 45643 xlimconst2 45850 cncfiooicclem1 45908 stoweidlem28 46043 nfdfat 47139 setrec2lem2 49213 setrec2 49214 |
| Copyright terms: Public domain | W3C validator |