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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 4904 | . 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) | |
2 | nffun.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | 2 | nfrel 4425 | . . 3 ⊢ Ⅎ𝑥Rel 𝐹 |
4 | 2 | nfcnv 4514 | . . . . 5 ⊢ Ⅎ𝑥◡𝐹 |
5 | 2, 4 | nfco 4501 | . . . 4 ⊢ Ⅎ𝑥(𝐹 ∘ ◡𝐹) |
6 | nfcv 2178 | . . . 4 ⊢ Ⅎ𝑥 I | |
7 | 5, 6 | nfss 2938 | . . 3 ⊢ Ⅎ𝑥(𝐹 ∘ ◡𝐹) ⊆ I |
8 | 3, 7 | nfan 1457 | . 2 ⊢ Ⅎ𝑥(Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ) |
9 | 1, 8 | nfxfr 1363 | 1 ⊢ Ⅎ𝑥Fun 𝐹 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 97 Ⅎwnf 1349 Ⅎwnfc 2165 ⊆ wss 2917 I cid 4025 ◡ccnv 4344 ∘ ccom 4349 Rel wrel 4350 Fun wfun 4896 |
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-fun 4904 |
This theorem is referenced by: nffn 4995 nff1 5090 fliftfun 5436 |
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