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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 3459 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
2 | 1 | ralimi 3128 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V) |
3 | mptfnf.0 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | mptfnf 6455 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
5 | 2, 4 | sylib 221 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 Ⅎwnfc 2936 ∀wral 3106 Vcvv 3441 ↦ cmpt 5110 Fn wfn 6319 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-v 3443 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-fun 6326 df-fn 6327 |
This theorem is referenced by: offval2f 7401 esumgsum 31414 esumc 31420 bj-mptval 34532 rfovcnvf1od 40705 dssmapf1od 40722 ntrrn 40825 dssmapntrcls 40831 |
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