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Mirrors > Home > MPE Home > Th. List > nffn | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for a function with domain. (Contributed by NM, 30-Jan-2004.) |
Ref | Expression |
---|---|
nffn.1 | ⊢ Ⅎ𝑥𝐹 |
nffn.2 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nffn | ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fn 5807 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
2 | nffn.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | 2 | nffun 5826 | . . 3 ⊢ Ⅎ𝑥Fun 𝐹 |
4 | 2 | nfdm 5288 | . . . 4 ⊢ Ⅎ𝑥dom 𝐹 |
5 | nffn.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
6 | 4, 5 | nfeq 2762 | . . 3 ⊢ Ⅎ𝑥dom 𝐹 = 𝐴 |
7 | 3, 6 | nfan 1816 | . 2 ⊢ Ⅎ𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴) |
8 | 1, 7 | nfxfr 1771 | 1 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 Ⅎwnf 1699 Ⅎwnfc 2738 dom cdm 5038 Fun wfun 5798 Fn wfn 5799 |
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-fun 5806 df-fn 5807 |
This theorem is referenced by: nff 5954 nffo 6027 feqmptdf 6161 nfixp 7813 nfixp1 7814 bnj1463 30377 choicefi 38387 stoweidlem31 38924 stoweidlem35 38928 stoweidlem59 38952 |
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