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Mirrors > Home > ILE Home > Th. List > dfmpt2 | GIF version |
Description: Alternate definition for the "maps to" notation df-mpt2 5517 (although it requires that 𝐶 be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
dfmpt2.1 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
dfmpt2 | ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {〈〈𝑥, 𝑦〉, 𝐶〉} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpt2mpts 5824 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶) | |
2 | vex 2560 | . . . . 5 ⊢ 𝑤 ∈ V | |
3 | 1stexg 5794 | . . . . 5 ⊢ (𝑤 ∈ V → (1st ‘𝑤) ∈ V) | |
4 | 2, 3 | ax-mp 7 | . . . 4 ⊢ (1st ‘𝑤) ∈ V |
5 | 2ndexg 5795 | . . . . . 6 ⊢ (𝑤 ∈ V → (2nd ‘𝑤) ∈ V) | |
6 | 2, 5 | ax-mp 7 | . . . . 5 ⊢ (2nd ‘𝑤) ∈ V |
7 | dfmpt2.1 | . . . . 5 ⊢ 𝐶 ∈ V | |
8 | 6, 7 | csbexa 3886 | . . . 4 ⊢ ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V |
9 | 4, 8 | csbexa 3886 | . . 3 ⊢ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V |
10 | 9 | dfmpt 5340 | . 2 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶) = ∪ 𝑤 ∈ (𝐴 × 𝐵){〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} |
11 | nfcv 2178 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
12 | nfcsb1v 2882 | . . . . 5 ⊢ Ⅎ𝑥⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 | |
13 | 11, 12 | nfop 3565 | . . . 4 ⊢ Ⅎ𝑥〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉 |
14 | 13 | nfsn 3430 | . . 3 ⊢ Ⅎ𝑥{〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} |
15 | nfcv 2178 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
16 | nfcv 2178 | . . . . . 6 ⊢ Ⅎ𝑦(1st ‘𝑤) | |
17 | nfcsb1v 2882 | . . . . . 6 ⊢ Ⅎ𝑦⦋(2nd ‘𝑤) / 𝑦⦌𝐶 | |
18 | 16, 17 | nfcsb 2884 | . . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 |
19 | 15, 18 | nfop 3565 | . . . 4 ⊢ Ⅎ𝑦〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉 |
20 | 19 | nfsn 3430 | . . 3 ⊢ Ⅎ𝑦{〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} |
21 | nfcv 2178 | . . 3 ⊢ Ⅎ𝑤{〈〈𝑥, 𝑦〉, 𝐶〉} | |
22 | id 19 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → 𝑤 = 〈𝑥, 𝑦〉) | |
23 | csbopeq1a 5814 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 = 𝐶) | |
24 | 22, 23 | opeq12d 3557 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → 〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉 = 〈〈𝑥, 𝑦〉, 𝐶〉) |
25 | 24 | sneqd 3388 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → {〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} = {〈〈𝑥, 𝑦〉, 𝐶〉}) |
26 | 14, 20, 21, 25 | iunxpf 4484 | . 2 ⊢ ∪ 𝑤 ∈ (𝐴 × 𝐵){〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {〈〈𝑥, 𝑦〉, 𝐶〉} |
27 | 1, 10, 26 | 3eqtri 2064 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {〈〈𝑥, 𝑦〉, 𝐶〉} |
Colors of variables: wff set class |
Syntax hints: = wceq 1243 ∈ wcel 1393 Vcvv 2557 ⦋csb 2852 {csn 3375 〈cop 3378 ∪ ciun 3657 ↦ cmpt 3818 × cxp 4343 ‘cfv 4902 ↦ cmpt2 5514 1st c1st 5765 2nd c2nd 5766 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-un 4170 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-reu 2313 df-v 2559 df-sbc 2765 df-csb 2853 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-id 4030 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-oprab 5516 df-mpt2 5517 df-1st 5767 df-2nd 5768 |
This theorem is referenced by: (None) |
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