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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 5808 | . 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
2 | nff.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | nff.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
4 | 2, 3 | nffn 5901 | . . 3 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
5 | 2 | nfrn 5289 | . . . 4 ⊢ Ⅎ𝑥ran 𝐹 |
6 | nff.3 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
7 | 5, 6 | nfss 3561 | . . 3 ⊢ Ⅎ𝑥ran 𝐹 ⊆ 𝐵 |
8 | 4, 7 | nfan 1816 | . 2 ⊢ Ⅎ𝑥(𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵) |
9 | 1, 8 | nfxfr 1771 | 1 ⊢ Ⅎ𝑥 𝐹:𝐴⟶𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 Ⅎwnf 1699 Ⅎwnfc 2738 ⊆ wss 3540 ran crn 5039 Fn wfn 5799 ⟶wf 5800 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-fun 5806 df-fn 5807 df-f 5808 |
This theorem is referenced by: nff1 6012 nfwrd 13188 lfgrnloop 25791 fcomptf 28840 aciunf1lem 28844 esumfzf 29458 esumfsup 29459 poimirlem24 32603 sdclem1 32709 dffo3f 38359 fmuldfeqlem1 38649 fnlimfvre 38741 dvnmul 38833 stoweidlem53 38946 stoweidlem54 38947 stoweidlem57 38950 sge0iunmpt 39311 |
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