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Theorem mpofun 7269
 Description: The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.)
Hypothesis
Ref Expression
mpofun.1 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
Assertion
Ref Expression
mpofun 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 mpofun
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqtr3 2847 . . . . . 6 ((𝑧 = 𝐶𝑤 = 𝐶) → 𝑧 = 𝑤)
21ad2ant2l 742 . . . . 5 ((((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤)
32gen2 1790 . . . 4 𝑧𝑤((((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤)
4 eqeq1 2829 . . . . . 6 (𝑧 = 𝑤 → (𝑧 = 𝐶𝑤 = 𝐶))
54anbi2d 628 . . . . 5 (𝑧 = 𝑤 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)))
65mo4 2647 . . . 4 (∃*𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ∀𝑧𝑤((((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤))
73, 6mpbir 232 . . 3 ∃*𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)
87funoprab 7267 . 2 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
9 mpofun.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
10 df-mpo 7156 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
119, 10eqtri 2848 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
1211funeqi 6372 . 2 (Fun 𝐹 ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)})
138, 12mpbir 232 1 Fun 𝐹
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1528   = wceq 1530   ∈ wcel 2107  ∃*wmo 2617  Fun wfun 6345  {coprab 7152   ∈ cmpo 7153 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-fun 6353  df-oprab 7155  df-mpo 7156 This theorem is referenced by:  ofexg  7406  mpoexxg  7767  mpoexw  7770  mpocurryd  7929  imasvscafn  16802  coapm  17323  oppglsm  18689  gsum2d2lem  19015  evlslem2  20212  xkococnlem  22183  ucnima  22805  ucnprima  22806  fmucnd  22816  smatrcl  30947  smatlem  30948  txomap  30984  tpr2rico  31041  elunirnmbfm  31397  scutf  33157  relowlpssretop  34514  aovmpt4g  43262  mpoexxg2  44214
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