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| Description: Alternate definition for the maps-to notation df-mpo 7437 (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 8091 | . 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 5310 | . . . 4 ⊢ ⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V | 
| 4 | 3 | csbex 5310 | . . 3 ⊢ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 ∈ V | 
| 5 | 4 | dfmpt 7163 | . 2 ⊢ (𝑤 ∈ (𝐴 × 𝐵) ↦ ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶) = ∪ 𝑤 ∈ (𝐴 × 𝐵){〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} | 
| 6 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑥𝑤 | |
| 7 | nfcsb1v 3922 | . . . . 5 ⊢ Ⅎ𝑥⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 | |
| 8 | 6, 7 | nfop 4888 | . . . 4 ⊢ Ⅎ𝑥〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉 | 
| 9 | 8 | nfsn 4706 | . . 3 ⊢ Ⅎ𝑥{〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} | 
| 10 | nfcv 2904 | . . . . 5 ⊢ Ⅎ𝑦𝑤 | |
| 11 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑦(1st ‘𝑤) | |
| 12 | nfcsb1v 3922 | . . . . . 6 ⊢ Ⅎ𝑦⦋(2nd ‘𝑤) / 𝑦⦌𝐶 | |
| 13 | 11, 12 | nfcsbw 3924 | . . . . 5 ⊢ Ⅎ𝑦⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 | 
| 14 | 10, 13 | nfop 4888 | . . . 4 ⊢ Ⅎ𝑦〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉 | 
| 15 | 14 | nfsn 4706 | . . 3 ⊢ Ⅎ𝑦{〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} | 
| 16 | nfcv 2904 | . . 3 ⊢ Ⅎ𝑤{〈〈𝑥, 𝑦〉, 𝐶〉} | |
| 17 | id 22 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → 𝑤 = 〈𝑥, 𝑦〉) | |
| 18 | csbopeq1a 8076 | . . . . 5 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶 = 𝐶) | |
| 19 | 17, 18 | opeq12d 4880 | . . . 4 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → 〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉 = 〈〈𝑥, 𝑦〉, 𝐶〉) | 
| 20 | 19 | sneqd 4637 | . . 3 ⊢ (𝑤 = 〈𝑥, 𝑦〉 → {〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} = {〈〈𝑥, 𝑦〉, 𝐶〉}) | 
| 21 | 9, 15, 16, 20 | iunxpf 5858 | . 2 ⊢ ∪ 𝑤 ∈ (𝐴 × 𝐵){〈𝑤, ⦋(1st ‘𝑤) / 𝑥⦌⦋(2nd ‘𝑤) / 𝑦⦌𝐶〉} = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {〈〈𝑥, 𝑦〉, 𝐶〉} | 
| 22 | 1, 5, 21 | 3eqtri 2768 | 1 ⊢ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 {〈〈𝑥, 𝑦〉, 𝐶〉} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∈ wcel 2107 Vcvv 3479 ⦋csb 3898 {csn 4625 〈cop 4631 ∪ ciun 4990 ↦ cmpt 5224 × cxp 5682 ‘cfv 6560 ∈ cmpo 7434 1st c1st 8013 2nd c2nd 8014 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-oprab 7436 df-mpo 7437 df-1st 8015 df-2nd 8016 | 
| This theorem is referenced by: fpar 8142 | 
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