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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 6565 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
| 2 | nff.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
| 3 | nff.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 4 | 2, 3 | nffn 6667 | . . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
| 5 | 2 | nfrn 5963 | . . . 4 ⊢ Ⅎ𝑥ran 𝐹 |
| 6 | nff.3 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
| 7 | 5, 6 | nfss 3976 | . . 3 ⊢ Ⅎ𝑥ran 𝐹 ⊆ 𝐵 |
| 8 | 4, 7 | nfan 1899 | . 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) |
| 9 | 1, 8 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 Ⅎwnf 1783 Ⅎwnfc 2890 ⊆ wss 3951 ran crn 5686 Fn wfn 6556 ⟶wf 6557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 |
| This theorem is referenced by: nff1 6802 dffo3f 7126 nfwrd 14581 lfgrnloop 29142 fcomptf 32668 aciunf1lem 32672 fnpreimac 32681 esumfzf 34070 esumfsup 34071 poimirlem24 37651 sdclem1 37750 nfrelp 44970 fmuldfeqlem1 45597 fnlimfvre 45689 dvnmul 45958 stoweidlem53 46068 stoweidlem54 46069 stoweidlem57 46072 sge0iunmpt 46433 |
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