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Mirrors > Home > MPE Home > Th. List > fnmptf | Structured version Visualization version GIF version |
Description: The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.) |
Ref | Expression |
---|---|
mptfnf.0 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
fnmptf | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 3429 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
2 | 1 | ralimi 3161 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V) |
3 | mptfnf.0 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | mptfnf 6252 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
5 | 2, 4 | sylib 210 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2164 Ⅎwnfc 2956 ∀wral 3117 Vcvv 3414 ↦ cmpt 4954 Fn wfn 6122 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-fun 6129 df-fn 6130 |
This theorem is referenced by: offval2f 7174 esumgsum 30648 esumc 30654 bj-mptval 33592 rfovcnvf1od 39137 dssmapf1od 39154 ntrrn 39259 dssmapntrcls 39265 |
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