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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 6560 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | nff.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | nff.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nffn 6661 | . . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
5 | 2 | nfrn 5960 | . . . 4 ⊢ Ⅎ𝑥ran 𝐹 |
6 | nff.3 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 5, 6 | nfss 3972 | . . 3 ⊢ Ⅎ𝑥ran 𝐹 ⊆ 𝐵 |
8 | 4, 7 | nfan 1895 | . 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) |
9 | 1, 8 | nfxfr 1848 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 394 Ⅎwnf 1778 Ⅎwnfc 2876 ⊆ wss 3947 ran crn 5685 Fn wfn 6551 ⟶wf 6552 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ral 3052 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-ss 3964 df-nul 4326 df-if 4534 df-sn 4634 df-pr 4636 df-op 4640 df-br 5156 df-opab 5218 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-fun 6558 df-fn 6559 df-f 6560 |
This theorem is referenced by: nff1 6798 dffo3f 7122 nfwrd 14553 lfgrnloop 29064 fcomptf 32577 aciunf1lem 32581 fnpreimac 32590 esumfzf 33904 esumfsup 33905 poimirlem24 37347 sdclem1 37446 fmuldfeqlem1 45221 fnlimfvre 45313 dvnmul 45582 stoweidlem53 45692 stoweidlem54 45693 stoweidlem57 45696 sge0iunmpt 46057 |
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