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Mirrors > Home > ILE Home > Th. List > dfmptg | GIF version |
Description: Alternate definition for the maps-to notation df-mpt 3961 (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 5215 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) | |
2 | vex 2663 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | xpsng 5563 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐵 ∈ 𝑉) → ({𝑥} × {𝐵}) = {〈𝑥, 𝐵〉}) | |
4 | 2, 3 | mpan 420 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → ({𝑥} × {𝐵}) = {〈𝑥, 𝐵〉}) |
5 | 4 | ralimi 2472 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = {〈𝑥, 𝐵〉}) |
6 | iuneq2 3799 | . . 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 | syl5eq 2162 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∈ wcel 1465 ∀wral 2393 Vcvv 2660 {csn 3497 〈cop 3500 ∪ ciun 3783 ↦ cmpt 3959 × cxp 4507 |
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 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-v 2662 df-sbc 2883 df-csb 2976 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-id 4185 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 |
This theorem is referenced by: fnasrng 5568 |
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