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Mirrors > Home > MPE Home > Th. List > nff | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for a mapping. (Contributed by NM, 29-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.) |
Ref | Expression |
---|---|
nff.1 | ⊢ Ⅎ𝑥𝐹 |
nff.2 | ⊢ Ⅎ𝑥𝐴 |
nff.3 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nff | ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-f 6328 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | nff.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | nff.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nffn 6422 | . . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
5 | 2 | nfrn 5788 | . . . 4 ⊢ Ⅎ𝑥ran 𝐹 |
6 | nff.3 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 5, 6 | nfss 3907 | . . 3 ⊢ Ⅎ𝑥ran 𝐹 ⊆ 𝐵 |
8 | 4, 7 | nfan 1900 | . 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) |
9 | 1, 8 | nfxfr 1854 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 Ⅎwnf 1785 Ⅎwnfc 2936 ⊆ wss 3881 ran crn 5520 Fn wfn 6319 ⟶wf 6320 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-fun 6326 df-fn 6327 df-f 6328 |
This theorem is referenced by: nff1 6547 nfwrd 13886 lfgrnloop 26918 fcomptf 30421 aciunf1lem 30425 fnpreimac 30434 esumfzf 31438 esumfsup 31439 poimirlem24 35081 sdclem1 35181 dffo3f 41806 fmuldfeqlem1 42224 fnlimfvre 42316 dvnmul 42585 stoweidlem53 42695 stoweidlem54 42696 stoweidlem57 42699 sge0iunmpt 43057 |
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