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Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nfdfat | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for "defined at". To prove a deduction version of this theorem is not easily possible because many deduction versions for bound-variable hypothesis builder for constructs the definition of "defined at" is based on are not available (e.g., for Fun/Rel, dom, ⊆, etc.). (Contributed by Alexander van der Vekens, 26-May-2017.) |
Ref | Expression |
---|---|
nfdfat.1 | ⊢ Ⅎ𝑥𝐹 |
nfdfat.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfdfat | ⊢ Ⅎ𝑥 𝐹 defAt 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dfat 47026 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
2 | nfdfat.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfdfat.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
4 | 3 | nfdm 5960 | . . . 4 ⊢ Ⅎ𝑥dom 𝐹 |
5 | 2, 4 | nfel 2916 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ dom 𝐹 |
6 | 2 | nfsn 4715 | . . . . 5 ⊢ Ⅎ𝑥{𝐴} |
7 | 3, 6 | nfres 5997 | . . . 4 ⊢ Ⅎ𝑥(𝐹 ↾ {𝐴}) |
8 | 7 | nffun 6587 | . . 3 ⊢ Ⅎ𝑥Fun (𝐹 ↾ {𝐴}) |
9 | 5, 8 | nfan 1895 | . 2 ⊢ Ⅎ𝑥(𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) |
10 | 1, 9 | nfxfr 1848 | 1 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 Ⅎwnf 1778 ∈ wcel 2104 Ⅎwnfc 2886 {csn 4631 dom cdm 5684 ↾ cres 5686 Fun wfun 6553 defAt wdfat 47023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ral 3058 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-br 5151 df-opab 5213 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-res 5696 df-fun 6561 df-dfat 47026 |
This theorem is referenced by: nfafv 47043 nfafv2 47125 |
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