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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 415 | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉)) |
4 | elex 3512 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → 𝐵 ∈ V) | |
5 | 3, 4 | syl6 35 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ V)) |
6 | 1, 5 | ralrimi 3216 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ V) |
7 | mptfnd.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
8 | 7 | mptfnf 6483 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
9 | 6, 8 | sylib 220 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 Ⅎwnf 1784 ∈ wcel 2114 Ⅎwnfc 2961 ∀wral 3138 Vcvv 3494 ↦ cmpt 5146 Fn wfn 6350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-fun 6357 df-fn 6358 |
This theorem is referenced by: smflimsuplem2 43115 |
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