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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 3345 | . . 3 ⊢ (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵) | |
2 | 1 | ralbii 2963 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵) |
3 | mptfng.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | df-mpt 4645 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
5 | 3, 4 | eqtri 2632 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
6 | 5 | fnopabg 5930 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵 ↔ 𝐹 Fn 𝐴) |
7 | 2, 6 | bitri 263 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∃!weu 2458 ∀wral 2896 Vcvv 3173 {copab 4642 ↦ cmpt 4643 Fn wfn 5799 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-fun 5806 df-fn 5807 |
This theorem is referenced by: fnmpt 5933 fnmpti 5935 mpteqb 6207 ofmpteq 6814 bdayfo 31074 fobigcup 31177 dihf11lem 35573 |
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