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Theorem dfmpt2 7418
Description: Alternate definition for the "maps to" notation df-mpt2 6798 (although it requires that 𝐶 be a set). (Contributed by NM, 19-Dec-2008.) (Revised by Mario Carneiro, 31-Aug-2015.)
Hypothesis
Ref Expression
dfmpt2.1 𝐶 ∈ V
Assertion
Ref Expression
dfmpt2 (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 dfmpt2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 mpt2mpts 7384 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑤 ∈ (𝐴 × 𝐵) ↦ (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶)
2 dfmpt2.1 . . . . 5 𝐶 ∈ V
32csbex 4927 . . . 4 (2nd𝑤) / 𝑦𝐶 ∈ V
43csbex 4927 . . 3 (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶 ∈ V
54dfmpt 6553 . 2 (𝑤 ∈ (𝐴 × 𝐵) ↦ (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶) = 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩}
6 nfcv 2913 . . . . 5 𝑥𝑤
7 nfcsb1v 3698 . . . . 5 𝑥(1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
86, 7nfop 4555 . . . 4 𝑥𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
98nfsn 4379 . . 3 𝑥{⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩}
10 nfcv 2913 . . . . 5 𝑦𝑤
11 nfcv 2913 . . . . . 6 𝑦(1st𝑤)
12 nfcsb1v 3698 . . . . . 6 𝑦(2nd𝑤) / 𝑦𝐶
1311, 12nfcsb 3700 . . . . 5 𝑦(1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
1410, 13nfop 4555 . . . 4 𝑦𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
1514nfsn 4379 . . 3 𝑦{⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩}
16 nfcv 2913 . . 3 𝑤{⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
17 id 22 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → 𝑤 = ⟨𝑥, 𝑦⟩)
18 csbopeq1a 7370 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶 = 𝐶)
1917, 18opeq12d 4547 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩ = ⟨⟨𝑥, 𝑦⟩, 𝐶⟩)
2019sneqd 4328 . . 3 (𝑤 = ⟨𝑥, 𝑦⟩ → {⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩} = {⟨⟨𝑥, 𝑦⟩, 𝐶⟩})
219, 15, 16, 20iunxpf 5409 . 2 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩} = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
221, 5, 213eqtri 2797 1 (𝑥𝐴, 𝑦𝐵𝐶) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  Vcvv 3351  csb 3682  {csn 4316  cop 4322   ciun 4654  cmpt 4863   × cxp 5247  cfv 6031  cmpt2 6795  1st c1st 7313  2nd c2nd 7314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316
This theorem is referenced by:  fpar  7432
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