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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 47131 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
| 2 | nfdfat.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 3 | nfdfat.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
| 4 | 3 | nfdm 5962 | . . . 4 ⊢ Ⅎ𝑥dom 𝐹 |
| 5 | 2, 4 | nfel 2920 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ dom 𝐹 |
| 6 | 2 | nfsn 4707 | . . . . 5 ⊢ Ⅎ𝑥{𝐴} |
| 7 | 3, 6 | nfres 5999 | . . . 4 ⊢ Ⅎ𝑥(𝐹 ↾ {𝐴}) |
| 8 | 7 | nffun 6589 | . . 3 ⊢ Ⅎ𝑥Fun (𝐹 ↾ {𝐴}) |
| 9 | 5, 8 | nfan 1899 | . 2 ⊢ Ⅎ𝑥(𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) |
| 10 | 1, 9 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 {csn 4626 dom cdm 5685 ↾ cres 5687 Fun wfun 6555 defAt wdfat 47128 |
| 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-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-res 5697 df-fun 6563 df-dfat 47131 |
| This theorem is referenced by: nfafv 47148 nfafv2 47230 |
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