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Mirrors > Home > ILE Home > Th. List > dfmpo | GIF version |
Description: Alternate definition for the maps-to notation df-mpo 5779 (although it requires that 𝐶 be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
dfmpo.1 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
dfmpo | ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {〈〈𝑥, 𝑦〉, 𝐶〉} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpompts 6096 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶) | |
2 | vex 2689 | . . . . 5 ⊢ 𝑤 ∈ V | |
3 | 1stexg 6065 | . . . . 5 ⊢ (𝑤 ∈ V → (1st ‘𝑤) ∈ V) | |
4 | 2, 3 | ax-mp 5 | . . . 4 ⊢ (1st ‘𝑤) ∈ V |
5 | 2ndexg 6066 | . . . . . 6 ⊢ (𝑤 ∈ V → (2nd ‘𝑤) ∈ V) | |
6 | 2, 5 | ax-mp 5 | . . . . 5 ⊢ (2nd ‘𝑤) ∈ V |
7 | dfmpo.1 | . . . . 5 ⊢ 𝐶 ∈ V | |
8 | 6, 7 | csbexa 4057 | . . . 4 ⊢ ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V |
9 | 4, 8 | csbexa 4057 | . . 3 ⊢ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V |
10 | 9 | dfmpt 5597 | . 2 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶) = ∪ 𝑤 ∈ (𝐴 × 𝐵){〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} |
11 | nfcv 2281 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
12 | nfcsb1v 3035 | . . . . 5 ⊢ Ⅎ𝑥⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 | |
13 | 11, 12 | nfop 3721 | . . . 4 ⊢ Ⅎ𝑥〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉 |
14 | 13 | nfsn 3583 | . . 3 ⊢ Ⅎ𝑥{〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} |
15 | nfcv 2281 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
16 | nfcv 2281 | . . . . . 6 ⊢ Ⅎ𝑦(1st ‘𝑤) | |
17 | nfcsb1v 3035 | . . . . . 6 ⊢ Ⅎ𝑦⦋(2nd ‘𝑤) / 𝑦⦌𝐶 | |
18 | 16, 17 | nfcsb 3037 | . . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 |
19 | 15, 18 | nfop 3721 | . . . 4 ⊢ Ⅎ𝑦〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉 |
20 | 19 | nfsn 3583 | . . 3 ⊢ Ⅎ𝑦{〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} |
21 | nfcv 2281 | . . 3 ⊢ Ⅎ𝑤{〈〈𝑥, 𝑦〉, 𝐶〉} | |
22 | id 19 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → 𝑤 = 〈𝑥, 𝑦〉) | |
23 | csbopeq1a 6086 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 = 𝐶) | |
24 | 22, 23 | opeq12d 3713 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → 〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉 = 〈〈𝑥, 𝑦〉, 𝐶〉) |
25 | 24 | sneqd 3540 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → {〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} = {〈〈𝑥, 𝑦〉, 𝐶〉}) |
26 | 14, 20, 21, 25 | iunxpf 4687 | . 2 ⊢ ∪ 𝑤 ∈ (𝐴 × 𝐵){〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {〈〈𝑥, 𝑦〉, 𝐶〉} |
27 | 1, 10, 26 | 3eqtri 2164 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {〈〈𝑥, 𝑦〉, 𝐶〉} |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 Vcvv 2686 ⦋csb 3003 {csn 3527 〈cop 3530 ∪ ciun 3813 ↦ cmpt 3989 × cxp 4537 ‘cfv 5123 ∈ cmpo 5776 1st c1st 6036 2nd c2nd 6037 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-reu 2423 df-v 2688 df-sbc 2910 df-csb 3004 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 |
This theorem is referenced by: (None) |
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