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Mirrors > Home > ILE Home > Th. List > dfmptg | GIF version |
Description: Alternate definition for the "maps to" notation df-mpt 3870 (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 5092 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) | |
2 | vex 2617 | . . . . 5 ⊢ 𝑥 ∈ V | |
3 | xpsng 5417 | . . . . 5 ⊢ ((𝑥 ∈ V ∧ 𝐵 ∈ 𝑉) → ({𝑥} × {𝐵}) = {〈𝑥, 𝐵〉}) | |
4 | 2, 3 | mpan 415 | . . . 4 ⊢ (𝐵 ∈ 𝑉 → ({𝑥} × {𝐵}) = {〈𝑥, 𝐵〉}) |
5 | 4 | ralimi 2434 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → ∀𝑥 ∈ 𝐴 ({𝑥} × {𝐵}) = {〈𝑥, 𝐵〉}) |
6 | iuneq2 3723 | . . 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 2129 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ 𝑥 ∈ 𝐴 {〈𝑥, 𝐵〉}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1287 ∈ wcel 1436 ∀wral 2355 Vcvv 2614 {csn 3425 〈cop 3428 ∪ ciun 3707 ↦ cmpt 3868 × cxp 4402 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-14 1448 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 ax-sep 3925 ax-pow 3977 ax-pr 4003 |
This theorem depends on definitions: df-bi 115 df-3an 924 df-tru 1290 df-nf 1393 df-sb 1690 df-eu 1948 df-mo 1949 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-ral 2360 df-rex 2361 df-reu 2362 df-v 2616 df-sbc 2829 df-csb 2922 df-un 2990 df-in 2992 df-ss 2999 df-pw 3411 df-sn 3431 df-pr 3432 df-op 3434 df-iun 3709 df-br 3815 df-opab 3869 df-mpt 3870 df-id 4087 df-xp 4410 df-rel 4411 df-cnv 4412 df-co 4413 df-dm 4414 df-rn 4415 df-fun 4974 df-fn 4975 df-f 4976 df-f1 4977 df-fo 4978 df-f1o 4979 |
This theorem is referenced by: fnasrng 5422 |
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