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Theorem mpt2fun 6660
 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 2631 . . . . . 6 ((𝑧 = 𝐶𝑤 = 𝐶) → 𝑧 = 𝑤)
21ad2ant2l 778 . . . . 5 ((((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤)
32gen2 1714 . . . 4 𝑧𝑤((((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤)
4 eqeq1 2614 . . . . . 6 (𝑧 = 𝑤 → (𝑧 = 𝐶𝑤 = 𝐶))
54anbi2d 736 . . . . 5 (𝑧 = 𝑤 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)))
65mo4 2505 . . . 4 (∃*𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ∀𝑧𝑤((((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤))
73, 6mpbir 220 . . 3 ∃*𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)
87funoprab 6658 . 2 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
9 mpt2fun.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
10 df-mpt2 6554 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
119, 10eqtri 2632 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
1211funeqi 5824 . 2 (Fun 𝐹 ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)})
138, 12mpbir 220 1 Fun 𝐹
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∃*wmo 2459  Fun wfun 5798  {coprab 6550   ↦ cmpt2 6551 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-fun 5806  df-oprab 6553  df-mpt2 6554 This theorem is referenced by:  ofexg  6799  mpt2exxg  7133  mpt2curryd  7282  imasvscafn  16020  coapm  16544  oppglsm  17880  gsum2d2lem  18195  evlslem2  19333  xkococnlem  21272  ucnima  21895  ucnprima  21896  fmucnd  21906  smatrcl  29190  smatlem  29191  txomap  29229  tpr2rico  29286  elunirnmbfm  29642  relowlpssretop  32388  aovmpt4g  39930  mpt2exxg2  41909
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