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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 3512 | . . 3 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
2 | 1 | ralimi 3160 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V) |
3 | mptfnf.0 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
4 | 3 | mptfnf 6477 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
5 | 2, 4 | sylib 220 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 Ⅎwnfc 2961 ∀wral 3138 Vcvv 3494 ↦ cmpt 5138 Fn wfn 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-fun 6351 df-fn 6352 |
This theorem is referenced by: offval2f 7415 esumgsum 31299 esumc 31305 bj-mptval 34403 rfovcnvf1od 40343 dssmapf1od 40360 ntrrn 40465 dssmapntrcls 40471 |
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