![]() |
Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > mptfnd | 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 |
---|---|
mptfnd.1 | ⊢ Ⅎ𝑥𝐴 |
mptfnd.2 | ⊢ Ⅎ𝑥𝜑 |
mptfnd.3 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
Ref | Expression |
---|---|
mptfnd | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptfnd.2 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | mptfnd.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
3 | 2 | ex 411 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉)) |
4 | elex 3490 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
5 | 3, 4 | syl6 35 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ V)) |
6 | 1, 5 | ralrimi 3250 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V) |
7 | mptfnd.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
8 | 7 | mptfnf 6693 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
9 | 6, 8 | sylib 217 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 Ⅎwnf 1777 ∈ wcel 2098 Ⅎwnfc 2878 ∀wral 3057 Vcvv 3471 ↦ cmpt 5233 Fn wfn 6546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5151 df-opab 5213 df-mpt 5234 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-fun 6553 df-fn 6554 |
This theorem is referenced by: smflimsuplem2 46211 |
Copyright terms: Public domain | W3C validator |