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| Mirrors > Home > MPE Home > Th. List > mptfng | Structured version Visualization version GIF version | ||
| Description: The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.) |
| Ref | Expression |
|---|---|
| mptfng.1 | ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) |
| Ref | Expression |
|---|---|
| mptfng | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eueq 3714 | . . 3 ⊢ (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵) | |
| 2 | 1 | ralbii 3093 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵) |
| 3 | mptfng.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 4 | df-mpt 5226 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
| 5 | 3, 4 | eqtri 2765 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
| 6 | 5 | fnopabg 6705 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵 ↔ 𝐹 Fn 𝐴) |
| 7 | 2, 6 | bitri 275 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∃!weu 2568 ∀wral 3061 Vcvv 3480 {copab 5205 ↦ cmpt 5225 Fn wfn 6556 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-fun 6563 df-fn 6564 |
| This theorem is referenced by: fnmpt 6708 fnmpti 6711 mpteqb 7035 ofmpteq 7720 bdayfo 27722 fobigcup 35901 dihf11lem 41268 |
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