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Theorem funopab4 4987
Description: A class of ordered pairs of values in the form used by df-mpt 3861 is a function. (Contributed by NM, 17-Feb-2013.)
Assertion
Ref Expression
funopab4 Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)}
Distinct variable groups:   𝑥,𝑦   𝑦,𝐴
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem funopab4
StepHypRef Expression
1 simpr 108 . . 3 ((𝜑𝑦 = 𝐴) → 𝑦 = 𝐴)
21ssopab2i 4060 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
3 funopabeq 4986 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
4 funss 4970 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} → (Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} → Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)}))
52, 3, 4mp2 16 1 Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)}
Colors of variables: wff set class
Syntax hints:  wa 102   = wceq 1285  wss 2982  {copab 3858  Fun wfun 4946
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 3916  ax-pow 3968  ax-pr 3992
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 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-opab 3860  df-id 4076  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-fun 4954
This theorem is referenced by:  funmpt  4988
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