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Theorem mpofun 7532
Description: The maps-to notation for an operation is always a function. (Contributed by Scott Fenton, 21-Mar-2012.) (Proof shortened by SN, 23-Jul-2024.)
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 variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 moeq 3704 . . . 4 ∃*𝑧 𝑧 = 𝐶
21moani 2548 . . 3 ∃*𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)
32funoprab 7530 . 2 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
4 mpofun.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
5 df-mpo 7414 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
64, 5eqtri 2761 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
76funeqi 6570 . 2 (Fun 𝐹 ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)})
83, 7mpbir 230 1 Fun 𝐹
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wcel 2107  Fun wfun 6538  {coprab 7410  cmpo 7411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-fun 6546  df-oprab 7413  df-mpo 7414
This theorem is referenced by:  ofexg  7675  mpoexxg  8062  mpoexw  8065  mpocurryd  8254  imasvscafn  17483  coapm  18021  oppglsm  19510  gsum2d2lem  19841  evlslem2  21642  xkococnlem  23163  ucnima  23786  ucnprima  23787  fmucnd  23797  scutf  27313  smatrcl  32776  smatlem  32777  txomap  32814  tpr2rico  32892  elunirnmbfm  33250  relowlpssretop  36245  aovmpt4g  45909  mpoexxg2  47013
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