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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 43312 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
2 | nfdfat.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfdfat.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
4 | 3 | nfdm 5817 | . . . 4 ⊢ Ⅎ𝑥dom 𝐹 |
5 | 2, 4 | nfel 2992 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ dom 𝐹 |
6 | 2 | nfsn 4636 | . . . . 5 ⊢ Ⅎ𝑥{𝐴} |
7 | 3, 6 | nfres 5849 | . . . 4 ⊢ Ⅎ𝑥(𝐹 ↾ {𝐴}) |
8 | 7 | nffun 6372 | . . 3 ⊢ Ⅎ𝑥Fun (𝐹 ↾ {𝐴}) |
9 | 5, 8 | nfan 1896 | . 2 ⊢ Ⅎ𝑥(𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) |
10 | 1, 9 | nfxfr 1849 | 1 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 Ⅎwnf 1780 ∈ wcel 2110 Ⅎwnfc 2961 {csn 4560 dom cdm 5549 ↾ cres 5551 Fun wfun 6343 defAt wdfat 43309 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-res 5561 df-fun 6351 df-dfat 43312 |
This theorem is referenced by: nfafv 43329 nfafv2 43411 |
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