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Mirrors > Home > MPE Home > Th. List > nffun | Structured version Visualization version 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 6335 | . 2 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I )) | |
2 | nffun.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | 2 | nfrel 5619 | . . 3 ⊢ Ⅎ𝑥Rel 𝐹 |
4 | 2 | nfcnv 5715 | . . . . 5 ⊢ Ⅎ𝑥◡𝐹 |
5 | 2, 4 | nfco 5702 | . . . 4 ⊢ Ⅎ𝑥(𝐹 ∘ ◡𝐹) |
6 | nfcv 2899 | . . . 4 ⊢ Ⅎ𝑥 I | |
7 | 5, 6 | nfss 3867 | . . 3 ⊢ Ⅎ𝑥(𝐹 ∘ ◡𝐹) ⊆ I |
8 | 3, 7 | nfan 1905 | . 2 ⊢ Ⅎ𝑥(Rel 𝐹 ∧ (𝐹 ∘ ◡𝐹) ⊆ I ) |
9 | 1, 8 | nfxfr 1859 | 1 ⊢ Ⅎ𝑥Fun 𝐹 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 Ⅎwnf 1790 Ⅎwnfc 2879 ⊆ wss 3841 I cid 5424 ◡ccnv 5518 ∘ ccom 5523 Rel wrel 5524 Fun wfun 6327 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-v 3399 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-nul 4210 df-if 4412 df-sn 4514 df-pr 4516 df-op 4520 df-br 5028 df-opab 5090 df-rel 5526 df-cnv 5527 df-co 5528 df-fun 6335 |
This theorem is referenced by: nffn 6431 nff1 6566 fliftfun 7072 funimass4f 30538 nfdfat 44136 |
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