![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dfmpo | Structured version Visualization version GIF version |
Description: Alternate definition for the maps-to notation df-mpo 7140 (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 7745 | . 2 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶) | |
2 | dfmpo.1 | . . . . 5 ⊢ 𝐶 ∈ V | |
3 | 2 | csbex 5179 | . . . 4 ⊢ ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V |
4 | 3 | csbex 5179 | . . 3 ⊢ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V |
5 | 4 | dfmpt 6883 | . 2 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶) = ∪ 𝑤 ∈ (𝐴 × 𝐵){〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} |
6 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
7 | nfcsb1v 3852 | . . . . 5 ⊢ Ⅎ𝑥⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 | |
8 | 6, 7 | nfop 4781 | . . . 4 ⊢ Ⅎ𝑥〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉 |
9 | 8 | nfsn 4603 | . . 3 ⊢ Ⅎ𝑥{〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} |
10 | nfcv 2955 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
11 | nfcv 2955 | . . . . . 6 ⊢ Ⅎ𝑦(1st ‘𝑤) | |
12 | nfcsb1v 3852 | . . . . . 6 ⊢ Ⅎ𝑦⦋(2nd ‘𝑤) / 𝑦⦌𝐶 | |
13 | 11, 12 | nfcsbw 3854 | . . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 |
14 | 10, 13 | nfop 4781 | . . . 4 ⊢ Ⅎ𝑦〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉 |
15 | 14 | nfsn 4603 | . . 3 ⊢ Ⅎ𝑦{〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} |
16 | nfcv 2955 | . . 3 ⊢ Ⅎ𝑤{〈〈𝑥, 𝑦〉, 𝐶〉} | |
17 | id 22 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → 𝑤 = 〈𝑥, 𝑦〉) | |
18 | csbopeq1a 7731 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 = 𝐶) | |
19 | 17, 18 | opeq12d 4773 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → 〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉 = 〈〈𝑥, 𝑦〉, 𝐶〉) |
20 | 19 | sneqd 4537 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → {〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} = {〈〈𝑥, 𝑦〉, 𝐶〉}) |
21 | 9, 15, 16, 20 | iunxpf 5683 | . 2 ⊢ ∪ 𝑤 ∈ (𝐴 × 𝐵){〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {〈〈𝑥, 𝑦〉, 𝐶〉} |
22 | 1, 5, 21 | 3eqtri 2825 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {〈〈𝑥, 𝑦〉, 𝐶〉} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 Vcvv 3441 ⦋csb 3828 {csn 4525 〈cop 4531 ∪ ciun 4881 ↦ cmpt 5110 × cxp 5517 ‘cfv 6324 ∈ cmpo 7137 1st c1st 7669 2nd c2nd 7670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 |
This theorem is referenced by: fpar 7794 |
Copyright terms: Public domain | W3C validator |