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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 5686 | . 2 ⊢ (𝐴 ↾ 𝐵) = (𝐴 ∩ (𝐵 × V)) | |
2 | nfres.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
3 | nfres.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
4 | nfcv 2904 | . . . 4 ⊢ Ⅎ𝑥V | |
5 | 3, 4 | nfxp 5707 | . . 3 ⊢ Ⅎ𝑥(𝐵 × V) |
6 | 2, 5 | nfin 4214 | . 2 ⊢ Ⅎ𝑥(𝐴 ∩ (𝐵 × V)) |
7 | 1, 6 | nfcxfr 2902 | 1 ⊢ Ⅎ𝑥(𝐴 ↾ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnfc 2884 Vcvv 3475 ∩ cin 3945 × cxp 5672 ↾ cres 5676 |
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 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-rab 3434 df-in 3953 df-opab 5209 df-xp 5680 df-res 5686 |
This theorem is referenced by: nfima 6064 nffrecs 8262 nfwrecsOLD 8296 frsucmpt 8432 frsucmptn 8433 nfoi 9504 prdsdsf 23854 prdsxmet 23856 limciun 25392 nosupbnd2 27198 noinfbnd2 27213 2ndresdju 31851 gsumpart 32184 bnj1446 33993 bnj1447 33994 bnj1448 33995 bnj1466 34001 bnj1467 34002 bnj1519 34013 bnj1520 34014 bnj1529 34018 feqresmptf 43864 limcperiod 44278 xlimconst2 44485 cncfiooicclem1 44543 stoweidlem28 44678 nfdfat 45769 setrec2lem2 47640 setrec2 47641 |
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