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Theorem funopabeq 5001
Description: A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)
Assertion
Ref Expression
funopabeq Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem funopabeq
StepHypRef Expression
1 funopab 5000 . 2 (Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} ↔ ∀𝑥∃*𝑦 𝑦 = 𝐴)
2 moeq 2778 . 2 ∃*𝑦 𝑦 = 𝐴
31, 2mpgbir 1383 1 Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
Colors of variables: wff set class
Syntax hints:   = wceq 1285  ∃*wmo 1944  {copab 3864  Fun wfun 4961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 3999
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-br 3812  df-opab 3866  df-id 4083  df-xp 4405  df-rel 4406  df-cnv 4407  df-co 4408  df-fun 4969
This theorem is referenced by:  funopab4  5002
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