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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 6495 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
2 | nffn.1 | . . . 4 ⊢ Ⅎ𝑥𝐹 | |
3 | 2 | nffun 6520 | . . 3 ⊢ Ⅎ𝑥Fun 𝐹 |
4 | 2 | nfdm 5903 | . . . 4 ⊢ Ⅎ𝑥dom 𝐹 |
5 | nffn.2 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
6 | 4, 5 | nfeq 2919 | . . 3 ⊢ Ⅎ𝑥dom 𝐹 = 𝐴 |
7 | 3, 6 | nfan 1903 | . 2 ⊢ Ⅎ𝑥(Fun 𝐹 ∧ dom 𝐹 = 𝐴) |
8 | 1, 7 | nfxfr 1856 | 1 ⊢ Ⅎ𝑥 𝐹 Fn 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 Ⅎwnf 1786 Ⅎwnfc 2886 dom cdm 5631 Fun wfun 6486 Fn wfn 6487 |
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 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-fun 6494 df-fn 6495 |
This theorem is referenced by: nff 6660 nffo 6751 feqmptdf 6908 nfixpw 8788 nfixp 8789 nfixp1 8790 bnj1463 33447 choicefi 43216 stoweidlem31 44063 stoweidlem35 44067 stoweidlem59 44091 |
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