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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 45827 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
2 | nfdfat.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfdfat.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
4 | 3 | nfdm 5951 | . . . 4 ⊢ Ⅎ𝑥dom 𝐹 |
5 | 2, 4 | nfel 2918 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ dom 𝐹 |
6 | 2 | nfsn 4712 | . . . . 5 ⊢ Ⅎ𝑥{𝐴} |
7 | 3, 6 | nfres 5984 | . . . 4 ⊢ Ⅎ𝑥(𝐹 ↾ {𝐴}) |
8 | 7 | nffun 6572 | . . 3 ⊢ Ⅎ𝑥Fun (𝐹 ↾ {𝐴}) |
9 | 5, 8 | nfan 1903 | . 2 ⊢ Ⅎ𝑥(𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) |
10 | 1, 9 | nfxfr 1856 | 1 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 Ⅎwnf 1786 ∈ wcel 2107 Ⅎwnfc 2884 {csn 4629 dom cdm 5677 ↾ cres 5679 Fun wfun 6538 defAt wdfat 45824 |
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-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3063 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-br 5150 df-opab 5212 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-res 5689 df-fun 6546 df-dfat 45827 |
This theorem is referenced by: nfafv 45844 nfafv2 45926 |
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