Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > nffun | GIF version |
Description: Bound-variable hypothesis builder for a function. (Contributed by NM, 30-Jan-2004.) |
Ref | Expression |
---|---|
nffun.1 | ⊢ Ⅎ𝑥𝐹 |
Ref | Expression |
---|---|
nffun | ⊢ Ⅎ𝑥Fun 𝐹 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fun 5125 | . 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) | |
2 | nffun.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | 2 | nfrel 4624 | . . 3 ⊢ Ⅎ𝑥Rel 𝐹 |
4 | 2 | nfcnv 4718 | . . . . 5 ⊢ Ⅎ𝑥◡𝐹 |
5 | 2, 4 | nfco 4704 | . . . 4 ⊢ Ⅎ𝑥(𝐹 ∘ ◡𝐹) |
6 | nfcv 2281 | . . . 4 ⊢ Ⅎ𝑥 I | |
7 | 5, 6 | nfss 3090 | . . 3 ⊢ Ⅎ𝑥(𝐹 ∘ ◡𝐹) ⊆ I |
8 | 3, 7 | nfan 1544 | . 2 ⊢ Ⅎ𝑥(Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ) |
9 | 1, 8 | nfxfr 1450 | 1 ⊢ Ⅎ𝑥Fun 𝐹 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 Ⅎwnf 1436 Ⅎwnfc 2268 ⊆ wss 3071 I cid 4210 ◡ccnv 4538 ∘ ccom 4543 Rel wrel 4544 Fun wfun 5117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-v 2688 df-un 3075 df-in 3077 df-ss 3084 df-sn 3533 df-pr 3534 df-op 3536 df-br 3930 df-opab 3990 df-rel 4546 df-cnv 4547 df-co 4548 df-fun 5125 |
This theorem is referenced by: nffn 5219 nff1 5326 fliftfun 5697 |
Copyright terms: Public domain | W3C validator |