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Theorem mpt2fun 5603
Description: The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)
Hypothesis
Ref Expression
mpt2fun.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
mpt2fun Fun 𝐹
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥,𝑦)   𝐹(𝑥,𝑦)

Proof of Theorem mpt2fun
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqtr3 2059 . . . . . 6 ((𝑧 = 𝐶𝑤 = 𝐶) → 𝑧 = 𝑤)
21ad2ant2l 477 . . . . 5 ((((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤)
32gen2 1339 . . . 4 𝑧𝑤((((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤)
4 eqeq1 2046 . . . . . 6 (𝑧 = 𝑤 → (𝑧 = 𝐶𝑤 = 𝐶))
54anbi2d 437 . . . . 5 (𝑧 = 𝑤 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)))
65mo4 1961 . . . 4 (∃*𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ∀𝑧𝑤((((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤))
73, 6mpbir 134 . . 3 ∃*𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)
87funoprab 5601 . 2 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
9 mpt2fun.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
10 df-mpt2 5517 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
119, 10eqtri 2060 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
1211funeqi 4922 . 2 (Fun 𝐹 ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)})
138, 12mpbir 134 1 Fun 𝐹
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97  wal 1241   = wceq 1243  wcel 1393  ∃*wmo 1901  Fun wfun 4896  {coprab 5513  cmpt2 5514
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-fun 4904  df-oprab 5516  df-mpt2 5517
This theorem is referenced by:  elmpt2cl  5698  ofexg  5716  mpt2exxg  5833  mpt2xopn0yelv  5854
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