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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 4905 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
2 | nffn.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | 2 | nffun 4924 | . . 3 ⊢ Ⅎ𝑥Fun 𝐹 |
4 | 2 | nfdm 4578 | . . . 4 ⊢ Ⅎ𝑥dom 𝐹 |
5 | nffn.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
6 | 4, 5 | nfeq 2185 | . . 3 ⊢ Ⅎ𝑥dom 𝐹 = 𝐴 |
7 | 3, 6 | nfan 1457 | . 2 ⊢ Ⅎ𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴) |
8 | 1, 7 | nfxfr 1363 | 1 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 = wceq 1243 Ⅎwnf 1349 Ⅎwnfc 2165 dom cdm 4345 Fun wfun 4896 Fn wfn 4897 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-sn 3381 df-pr 3382 df-op 3384 df-br 3765 df-opab 3819 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-fun 4904 df-fn 4905 |
This theorem is referenced by: nff 5043 nffo 5105 |
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