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Mirrors > Home > MPE Home > Th. List > dfmpo | Structured version Visualization version GIF version |
Description: Alternate definition for the maps-to notation df-mpo 7161 (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 7763 | . 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 5215 | . . . 4 ⊢ ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V |
4 | 3 | csbex 5215 | . . 3 ⊢ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V |
5 | 4 | dfmpt 6906 | . 2 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶) = ∪ 𝑤 ∈ (𝐴 × 𝐵){〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} |
6 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
7 | nfcsb1v 3907 | . . . . 5 ⊢ Ⅎ𝑥⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 | |
8 | 6, 7 | nfop 4819 | . . . 4 ⊢ Ⅎ𝑥〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉 |
9 | 8 | nfsn 4643 | . . 3 ⊢ Ⅎ𝑥{〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} |
10 | nfcv 2977 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
11 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑦(1st ‘𝑤) | |
12 | nfcsb1v 3907 | . . . . . 6 ⊢ Ⅎ𝑦⦋(2nd ‘𝑤) / 𝑦⦌𝐶 | |
13 | 11, 12 | nfcsbw 3909 | . . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 |
14 | 10, 13 | nfop 4819 | . . . 4 ⊢ Ⅎ𝑦〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉 |
15 | 14 | nfsn 4643 | . . 3 ⊢ Ⅎ𝑦{〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} |
16 | nfcv 2977 | . . 3 ⊢ Ⅎ𝑤{〈〈𝑥, 𝑦〉, 𝐶〉} | |
17 | id 22 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → 𝑤 = 〈𝑥, 𝑦〉) | |
18 | csbopeq1a 7749 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 = 𝐶) | |
19 | 17, 18 | opeq12d 4811 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → 〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉 = 〈〈𝑥, 𝑦〉, 𝐶〉) |
20 | 19 | sneqd 4579 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → {〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} = {〈〈𝑥, 𝑦〉, 𝐶〉}) |
21 | 9, 15, 16, 20 | iunxpf 5719 | . 2 ⊢ ∪ 𝑤 ∈ (𝐴 × 𝐵){〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {〈〈𝑥, 𝑦〉, 𝐶〉} |
22 | 1, 5, 21 | 3eqtri 2848 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {〈〈𝑥, 𝑦〉, 𝐶〉} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 Vcvv 3494 ⦋csb 3883 {csn 4567 〈cop 4573 ∪ ciun 4919 ↦ cmpt 5146 × cxp 5553 ‘cfv 6355 ∈ cmpo 7158 1st c1st 7687 2nd c2nd 7688 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 |
This theorem is referenced by: fpar 7811 |
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