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Mirrors > Home > MPE Home > Th. List > fnmptd | Structured version Visualization version GIF version |
Description: The maps-to notation defines a function with domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.) |
Ref | Expression |
---|---|
fnmptd.1 | ⊢ Ⅎ𝑥𝜑 |
fnmptd.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
fnmptd.3 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
Ref | Expression |
---|---|
fnmptd | ⊢ (𝜑 → 𝐹 Fn 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fnmptd.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | fnmptd.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
3 | 2 | ex 416 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝑉)) |
4 | 1, 3 | ralrimi 3129 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉) |
5 | fnmptd.3 | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
6 | 5 | fnmpt 6487 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → 𝐹 Fn 𝐴) |
7 | 4, 6 | syl 17 | 1 ⊢ (𝜑 → 𝐹 Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 Ⅎwnf 1790 ∈ wcel 2114 ∀wral 3054 ↦ cmpt 5120 Fn wfn 6344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ral 3059 df-v 3402 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-fun 6351 df-fn 6352 |
This theorem is referenced by: nsgmgc 31181 nsgqusf1o 31185 metakunt33 39788 limsupequzmptlem 42851 liminfval2 42891 smflimmpt 43922 smflimsuplem7 43938 cfsetsnfsetfo 44133 |
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