ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dfmpt2 GIF version

Theorem dfmpt2 5871
Description: Alternate definition for the "maps to" notation df-mpt2 5544 (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 5851 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑤 ∈ (𝐴 × 𝐵) ↦ (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶)
2 vex 2577 . . . . 5 𝑤 ∈ V
3 1stexg 5821 . . . . 5 (𝑤 ∈ V → (1st𝑤) ∈ V)
42, 3ax-mp 7 . . . 4 (1st𝑤) ∈ V
5 2ndexg 5822 . . . . . 6 (𝑤 ∈ V → (2nd𝑤) ∈ V)
62, 5ax-mp 7 . . . . 5 (2nd𝑤) ∈ V
7 dfmpt2.1 . . . . 5 𝐶 ∈ V
86, 7csbexa 3913 . . . 4 (2nd𝑤) / 𝑦𝐶 ∈ V
94, 8csbexa 3913 . . 3 (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶 ∈ V
109dfmpt 5367 . 2 (𝑤 ∈ (𝐴 × 𝐵) ↦ (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶) = 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩}
11 nfcv 2194 . . . . 5 𝑥𝑤
12 nfcsb1v 2909 . . . . 5 𝑥(1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
1311, 12nfop 3592 . . . 4 𝑥𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
1413nfsn 3457 . . 3 𝑥{⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩}
15 nfcv 2194 . . . . 5 𝑦𝑤
16 nfcv 2194 . . . . . 6 𝑦(1st𝑤)
17 nfcsb1v 2909 . . . . . 6 𝑦(2nd𝑤) / 𝑦𝐶
1816, 17nfcsb 2911 . . . . 5 𝑦(1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
1915, 18nfop 3592 . . . 4 𝑦𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
2019nfsn 3457 . . 3 𝑦{⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩}
21 nfcv 2194 . . 3 𝑤{⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
22 id 19 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → 𝑤 = ⟨𝑥, 𝑦⟩)
23 csbopeq1a 5841 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶 = 𝐶)
2422, 23opeq12d 3584 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩ = ⟨⟨𝑥, 𝑦⟩, 𝐶⟩)
2524sneqd 3415 . . 3 (𝑤 = ⟨𝑥, 𝑦⟩ → {⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩} = {⟨⟨𝑥, 𝑦⟩, 𝐶⟩})
2614, 20, 21, 25iunxpf 4511 . 2 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩} = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
271, 10, 263eqtri 2080 1 (𝑥𝐴, 𝑦𝐵𝐶) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
Colors of variables: wff set class
Syntax hints:   = wceq 1259  wcel 1409  Vcvv 2574  csb 2879  {csn 3402  cop 3405   ciun 3684  cmpt 3845   × cxp 4370  cfv 4929  cmpt2 5541  1st c1st 5792  2nd c2nd 5793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-reu 2330  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-f1 4934  df-fo 4935  df-f1o 4936  df-fv 4937  df-oprab 5543  df-mpt2 5544  df-1st 5794  df-2nd 5795
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator