MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mpt2fun Structured version   Visualization version   GIF version

Theorem mpt2fun 6964
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 2786 . . . . . 6 ((𝑧 = 𝐶𝑤 = 𝐶) → 𝑧 = 𝑤)
21ad2ant2l 752 . . . . 5 ((((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤)
32gen2 1891 . . . 4 𝑧𝑤((((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤)
4 eqeq1 2769 . . . . . 6 (𝑧 = 𝑤 → (𝑧 = 𝐶𝑤 = 𝐶))
54anbi2d 622 . . . . 5 (𝑧 = 𝑤 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)))
65mo4 2638 . . . 4 (∃*𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ∀𝑧𝑤((((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤))
73, 6mpbir 222 . . 3 ∃*𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)
87funoprab 6962 . 2 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
9 mpt2fun.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
10 df-mpt2 6851 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
119, 10eqtri 2787 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
1211funeqi 6091 . 2 (Fun 𝐹 ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)})
138, 12mpbir 222 1 Fun 𝐹
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1650   = wceq 1652  wcel 2155  ∃*wmo 2563  Fun wfun 6064  {coprab 6847  cmpt2 6848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-id 5187  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-fun 6072  df-oprab 6850  df-mpt2 6851
This theorem is referenced by:  ofexg  7103  mpt2exxg  7449  mpt2curryd  7602  imasvscafn  16477  coapm  17000  oppglsm  18335  gsum2d2lem  18652  evlslem2  19799  xkococnlem  21756  ucnima  22378  ucnprima  22379  fmucnd  22389  smatrcl  30330  smatlem  30331  txomap  30369  tpr2rico  30426  elunirnmbfm  30783  scutf  32384  relowlpssretop  33666  aovmpt4g  41973  mpt2exxg2  42809
  Copyright terms: Public domain W3C validator