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Theorem dfmpt2 6002
Description: Alternate definition for the maps-to notation df-mpt2 5671 (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 5982 . 2 (𝑥𝐴, 𝑦𝐵𝐶) = (𝑤 ∈ (𝐴 × 𝐵) ↦ (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶)
2 vex 2623 . . . . 5 𝑤 ∈ V
3 1stexg 5952 . . . . 5 (𝑤 ∈ V → (1st𝑤) ∈ V)
42, 3ax-mp 7 . . . 4 (1st𝑤) ∈ V
5 2ndexg 5953 . . . . . 6 (𝑤 ∈ V → (2nd𝑤) ∈ V)
62, 5ax-mp 7 . . . . 5 (2nd𝑤) ∈ V
7 dfmpt2.1 . . . . 5 𝐶 ∈ V
86, 7csbexa 3974 . . . 4 (2nd𝑤) / 𝑦𝐶 ∈ V
94, 8csbexa 3974 . . 3 (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶 ∈ V
109dfmpt 5488 . 2 (𝑤 ∈ (𝐴 × 𝐵) ↦ (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶) = 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩}
11 nfcv 2229 . . . . 5 𝑥𝑤
12 nfcsb1v 2964 . . . . 5 𝑥(1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
1311, 12nfop 3644 . . . 4 𝑥𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
1413nfsn 3506 . . 3 𝑥{⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩}
15 nfcv 2229 . . . . 5 𝑦𝑤
16 nfcv 2229 . . . . . 6 𝑦(1st𝑤)
17 nfcsb1v 2964 . . . . . 6 𝑦(2nd𝑤) / 𝑦𝐶
1816, 17nfcsb 2966 . . . . 5 𝑦(1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
1915, 18nfop 3644 . . . 4 𝑦𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶
2019nfsn 3506 . . 3 𝑦{⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩}
21 nfcv 2229 . . 3 𝑤{⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
22 id 19 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → 𝑤 = ⟨𝑥, 𝑦⟩)
23 csbopeq1a 5972 . . . . 5 (𝑤 = ⟨𝑥, 𝑦⟩ → (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶 = 𝐶)
2422, 23opeq12d 3636 . . . 4 (𝑤 = ⟨𝑥, 𝑦⟩ → ⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩ = ⟨⟨𝑥, 𝑦⟩, 𝐶⟩)
2524sneqd 3463 . . 3 (𝑤 = ⟨𝑥, 𝑦⟩ → {⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩} = {⟨⟨𝑥, 𝑦⟩, 𝐶⟩})
2614, 20, 21, 25iunxpf 4597 . 2 𝑤 ∈ (𝐴 × 𝐵){⟨𝑤, (1st𝑤) / 𝑥(2nd𝑤) / 𝑦𝐶⟩} = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
271, 10, 263eqtri 2113 1 (𝑥𝐴, 𝑦𝐵𝐶) = 𝑥𝐴 𝑦𝐵 {⟨⟨𝑥, 𝑦⟩, 𝐶⟩}
Colors of variables: wff set class
Syntax hints:   = wceq 1290  wcel 1439  Vcvv 2620  csb 2934  {csn 3450  cop 3453   ciun 3736  cmpt 3905   × cxp 4449  cfv 5028  cmpt2 5668  1st c1st 5923  2nd c2nd 5924
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-pow 4015  ax-pr 4045  ax-un 4269
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-rex 2366  df-reu 2367  df-v 2622  df-sbc 2842  df-csb 2935  df-un 3004  df-in 3006  df-ss 3013  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-iun 3738  df-br 3852  df-opab 3906  df-mpt 3907  df-id 4129  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-oprab 5670  df-mpt2 5671  df-1st 5925  df-2nd 5926
This theorem is referenced by: (None)
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