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Mirrors > Home > ILE Home > Th. List > mptfng | 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 2883 | . . 3 ⊢ (𝐵 ∈ V ↔ ∃!𝑦 𝑦 = 𝐵) | |
2 | 1 | ralbii 2463 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ ∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵) |
3 | mptfng.1 | . . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | df-mpt 4028 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} | |
5 | 3, 4 | eqtri 2178 | . . 3 ⊢ 𝐹 = {〈𝑥, 𝑦〉 ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} |
6 | 5 | fnopabg 5294 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∃!𝑦 𝑦 = 𝐵 ↔ 𝐹 Fn 𝐴) |
7 | 2, 6 | bitri 183 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ V ↔ 𝐹 Fn 𝐴) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 = wceq 1335 ∃!weu 2006 ∈ wcel 2128 ∀wral 2435 Vcvv 2712 {copab 4025 ↦ cmpt 4026 Fn wfn 5166 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-pow 4136 ax-pr 4170 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ral 2440 df-rex 2441 df-v 2714 df-un 3106 df-in 3108 df-ss 3115 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-br 3967 df-opab 4027 df-mpt 4028 df-id 4254 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-fun 5173 df-fn 5174 |
This theorem is referenced by: fnmpt 5297 fnmpti 5299 mpteqb 5559 cc3 7189 |
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