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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 46126 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
2 | nfdfat.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfdfat.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
4 | 3 | nfdm 5950 | . . . 4 ⊢ Ⅎ𝑥dom 𝐹 |
5 | 2, 4 | nfel 2917 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ dom 𝐹 |
6 | 2 | nfsn 4711 | . . . . 5 ⊢ Ⅎ𝑥{𝐴} |
7 | 3, 6 | nfres 5983 | . . . 4 ⊢ Ⅎ𝑥(𝐹 ↾ {𝐴}) |
8 | 7 | nffun 6571 | . . 3 ⊢ Ⅎ𝑥Fun (𝐹 ↾ {𝐴}) |
9 | 5, 8 | nfan 1902 | . 2 ⊢ Ⅎ𝑥(𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) |
10 | 1, 9 | nfxfr 1855 | 1 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2883 {csn 4628 dom cdm 5676 ↾ cres 5678 Fun wfun 6537 defAt wdfat 46123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ral 3062 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-res 5688 df-fun 6545 df-dfat 46126 |
This theorem is referenced by: nfafv 46143 nfafv2 46225 |
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