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Theorem dfmpo 8128
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.)
Hypothesis
Ref Expression
dfmpo.1 𝐶 ∈ V
Assertion
Ref Expression
dfmpo (𝑥𝐴, 𝑦𝐵𝐶) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦
Allowed substitution hints:   𝐶(𝑥,𝑦)

Proof of Theorem dfmpo
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 mpompts 8091 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑤 ∈ (𝐴 × 𝐵) ↦ (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶)
2 dfmpo.1 . . . . 5 𝐶 ∈ V
32csbex 5310 . . . 4 (2nd𝑤) / 𝑦𝐶 ∈ V
43csbex 5310 . . 3 (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶 ∈ V
54dfmpt 7163 . 2 (𝑤 ∈ (𝐴 × 𝐵) ↦ (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶) = 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩}
6 nfcv 2904 . . . . 5 𝑥𝑤
7 nfcsb1v 3922 . . . . 5 𝑥(1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
86, 7nfop 4888 . . . 4 𝑥𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
98nfsn 4706 . . 3 𝑥{⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩}
10 nfcv 2904 . . . . 5 𝑦𝑤
11 nfcv 2904 . . . . . 6 𝑦(1st𝑤)
12 nfcsb1v 3922 . . . . . 6 𝑦(2nd𝑤) / 𝑦𝐶
1311, 12nfcsbw 3924 . . . . 5 𝑦(1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
1410, 13nfop 4888 . . . 4 𝑦𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
1514nfsn 4706 . . 3 𝑦{⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩}
16 nfcv 2904 . . 3 𝑤{⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
17 id 22 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → 𝑤 = ⟨𝑥, 𝑦⟩)
18 csbopeq1a 8076 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶 = 𝐶)
1917, 18opeq12d 4880 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩ = ⟨⟨𝑥, 𝑦⟩, 𝐶⟩)
2019sneqd 4637 . . 3 (𝑤 = ⟨𝑥, 𝑦⟩ → {⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩} = {⟨⟨𝑥, 𝑦⟩, 𝐶⟩})
219, 15, 16, 20iunxpf 5858 . 2 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩} = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
221, 5, 213eqtri 2768 1 (𝑥𝐴, 𝑦𝐵𝐶) = 𝑥𝐴 𝑦𝐵 {⟨⟨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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