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Mirrors > Home > ILE Home > Th. List > dfmptg | GIF version |
Description: Alternate definition for the maps-to notation df-mpt 4092 (which requires that 𝐵 be a set). (Contributed by Jim Kingdon, 9-Jan-2019.) |
Ref | Expression |
---|---|
dfmptg | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfmpt3 5376 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) | |
2 | vex 2763 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | xpsng 5733 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐵 ∈ 𝑉) → ({𝑥} × {𝐵}) = {〈𝑥, 𝐵〉}) | |
4 | 2, 3 | mpan 424 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → ({𝑥} × {𝐵}) = {〈𝑥, 𝐵〉}) |
5 | 4 | ralimi 2557 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = {〈𝑥, 𝐵〉}) |
6 | iuneq2 3928 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = {〈𝑥, 𝐵〉} → ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉}) | |
7 | 5, 6 | syl 14 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉}) |
8 | 1, 7 | eqtrid 2238 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1364 ∈ wcel 2164 ∀wral 2472 Vcvv 2760 {csn 3618 〈cop 3621 ∪ ciun 3912 ↦ cmpt 4090 × cxp 4657 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2167 ax-ext 2175 ax-sep 4147 ax-pow 4203 ax-pr 4238 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-eu 2045 df-mo 2046 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ral 2477 df-rex 2478 df-reu 2479 df-v 2762 df-sbc 2986 df-csb 3081 df-un 3157 df-in 3159 df-ss 3166 df-pw 3603 df-sn 3624 df-pr 3625 df-op 3627 df-iun 3914 df-br 4030 df-opab 4091 df-mpt 4092 df-id 4324 df-xp 4665 df-rel 4666 df-cnv 4667 df-co 4668 df-dm 4669 df-rn 4670 df-fun 5256 df-fn 5257 df-f 5258 df-f1 5259 df-fo 5260 df-f1o 5261 |
This theorem is referenced by: fnasrng 5738 |
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