Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
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 43325 | . 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴}))) | |
2 | nfdfat.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfdfat.1 | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
4 | 3 | nfdm 5826 | . . . 4 ⊢ Ⅎ𝑥dom 𝐹 |
5 | 2, 4 | nfel 2995 | . . 3 ⊢ Ⅎ𝑥 𝐴 ∈ dom 𝐹 |
6 | 2 | nfsn 4646 | . . . . 5 ⊢ Ⅎ𝑥{𝐴} |
7 | 3, 6 | nfres 5858 | . . . 4 ⊢ Ⅎ𝑥(𝐹 ↾ {𝐴}) |
8 | 7 | nffun 6381 | . . 3 ⊢ Ⅎ𝑥Fun (𝐹 ↾ {𝐴}) |
9 | 5, 8 | nfan 1899 | . 2 ⊢ Ⅎ𝑥(𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) |
10 | 1, 9 | nfxfr 1852 | 1 ⊢ Ⅎ𝑥 𝐹 defAt 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 Ⅎwnf 1783 ∈ wcel 2113 Ⅎwnfc 2964 {csn 4570 dom cdm 5558 ↾ cres 5560 Fun wfun 6352 defAt wdfat 43322 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rab 3150 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-br 5070 df-opab 5132 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-res 5570 df-fun 6360 df-dfat 43325 |
This theorem is referenced by: nfafv 43342 nfafv2 43424 |
Copyright terms: Public domain | W3C validator |