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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 6564 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
| 2 | nffn.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
| 3 | 2 | nffun 6589 | . . 3 ⊢ Ⅎ𝑥Fun 𝐹 | 
| 4 | 2 | nfdm 5962 | . . . 4 ⊢ Ⅎ𝑥dom 𝐹 | 
| 5 | nffn.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 6 | 4, 5 | nfeq 2919 | . . 3 ⊢ Ⅎ𝑥dom 𝐹 = 𝐴 | 
| 7 | 3, 6 | nfan 1899 | . 2 ⊢ Ⅎ𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴) | 
| 8 | 1, 7 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ∧ wa 395 = wceq 1540 Ⅎwnf 1783 Ⅎwnfc 2890 dom cdm 5685 Fun wfun 6555 Fn wfn 6556 | 
| 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-fun 6563 df-fn 6564 | 
| This theorem is referenced by: nff 6732 nffo 6819 feqmptdf 6979 nfixpw 8956 nfixp 8957 nfixp1 8958 bnj1463 35069 choicefi 45205 stoweidlem31 46046 stoweidlem35 46050 stoweidlem59 46074 | 
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