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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 5184 | . 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) | |
2 | nffun.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | 2 | nfrel 4683 | . . 3 ⊢ Ⅎ𝑥Rel 𝐹 |
4 | 2 | nfcnv 4777 | . . . . 5 ⊢ Ⅎ𝑥◡𝐹 |
5 | 2, 4 | nfco 4763 | . . . 4 ⊢ Ⅎ𝑥(𝐹 ∘ ◡𝐹) |
6 | nfcv 2306 | . . . 4 ⊢ Ⅎ𝑥 I | |
7 | 5, 6 | nfss 3130 | . . 3 ⊢ Ⅎ𝑥(𝐹 ∘ ◡𝐹) ⊆ I |
8 | 3, 7 | nfan 1552 | . 2 ⊢ Ⅎ𝑥(Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ) |
9 | 1, 8 | nfxfr 1461 | 1 ⊢ Ⅎ𝑥Fun 𝐹 |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 Ⅎwnf 1447 Ⅎwnfc 2293 ⊆ wss 3111 I cid 4260 ◡ccnv 4597 ∘ ccom 4602 Rel wrel 4603 Fun wfun 5176 |
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 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-rel 4605 df-cnv 4606 df-co 4607 df-fun 5184 |
This theorem is referenced by: nffn 5278 nff1 5385 fliftfun 5758 |
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