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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 6498 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | nff.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | nff.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nffn 6599 | . . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
5 | 2 | nfrn 5906 | . . . 4 ⊢ Ⅎ𝑥ran 𝐹 |
6 | nff.3 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 5, 6 | nfss 3935 | . . 3 ⊢ Ⅎ𝑥ran 𝐹 ⊆ 𝐵 |
8 | 4, 7 | nfan 1903 | . 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) |
9 | 1, 8 | nfxfr 1856 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 Ⅎwnf 1786 Ⅎwnfc 2886 ⊆ wss 3909 ran crn 5633 Fn wfn 6489 ⟶wf 6490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ral 3064 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-fun 6496 df-fn 6497 df-f 6498 |
This theorem is referenced by: nff1 6734 nfwrd 14385 lfgrnloop 27905 fcomptf 31402 aciunf1lem 31406 fnpreimac 31415 esumfzf 32472 esumfsup 32473 poimirlem24 36034 sdclem1 36134 dffo3f 43297 fmuldfeqlem1 43718 fnlimfvre 43810 dvnmul 44079 stoweidlem53 44189 stoweidlem54 44190 stoweidlem57 44193 sge0iunmpt 44554 |
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