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Theorem mpofun 5839
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 2135 . . . . . 6 ((𝑧 = 𝐶𝑤 = 𝐶) → 𝑧 = 𝑤)
21ad2ant2l 497 . . . . 5 ((((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤)
32gen2 1409 . . . 4 𝑧𝑤((((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤)
4 eqeq1 2122 . . . . . 6 (𝑧 = 𝑤 → (𝑧 = 𝐶𝑤 = 𝐶))
54anbi2d 457 . . . . 5 (𝑧 = 𝑤 → (((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)))
65mo4 2036 . . . 4 (∃*𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ↔ ∀𝑧𝑤((((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶) ∧ ((𝑥𝐴𝑦𝐵) ∧ 𝑤 = 𝐶)) → 𝑧 = 𝑤))
73, 6mpbir 145 . . 3 ∃*𝑧((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)
87funoprab 5837 . 2 Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
9 mpofun.1 . . . 4 𝐹 = (𝑥𝐴, 𝑦𝐵𝐶)
10 df-mpo 5745 . . . 4 (𝑥𝐴, 𝑦𝐵𝐶) = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
119, 10eqtri 2136 . . 3 𝐹 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)}
1211funeqi 5112 . 2 (Fun 𝐹 ↔ Fun {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ ((𝑥𝐴𝑦𝐵) ∧ 𝑧 = 𝐶)})
138, 12mpbir 145 1 Fun 𝐹
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1312   = wceq 1314  wcel 1463  ∃*wmo 1976  Fun wfun 5085  {coprab 5741  cmpo 5742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-fun 5093  df-oprab 5744  df-mpo 5745
This theorem is referenced by:  elmpocl  5934  ofexg  5952  mpoexxg  6074  mpoxopn0yelv  6102
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