![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > nffn | GIF version |
Description: Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.) |
Ref | Expression |
---|---|
nffn.1 | ⊢ Ⅎ𝑥𝐹 |
nffn.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nffn | ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fn 5231 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
2 | nffn.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | 2 | nffun 5251 | . . 3 ⊢ Ⅎ𝑥Fun 𝐹 |
4 | 2 | nfdm 4883 | . . . 4 ⊢ Ⅎ𝑥dom 𝐹 |
5 | nffn.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
6 | 4, 5 | nfeq 2337 | . . 3 ⊢ Ⅎ𝑥dom 𝐹 = 𝐴 |
7 | 3, 6 | nfan 1575 | . 2 ⊢ Ⅎ𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴) |
8 | 1, 7 | nfxfr 1484 | 1 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 = wceq 1363 Ⅎwnf 1470 Ⅎwnfc 2316 dom cdm 4638 Fun wfun 5222 Fn wfn 5223 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-ext 2169 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-v 2751 df-un 3145 df-in 3147 df-ss 3154 df-sn 3610 df-pr 3611 df-op 3613 df-br 4016 df-opab 4077 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-fun 5230 df-fn 5231 |
This theorem is referenced by: nff 5374 nffo 5449 nfixpxy 6731 nfixp1 6732 |
Copyright terms: Public domain | W3C validator |