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Theorem funopab4 5130
Description: A class of ordered pairs of values in the form used by df-mpt 3961 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 109 . . 3 ((𝜑𝑦 = 𝐴) → 𝑦 = 𝐴)
21ssopab2i 4169 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
3 funopabeq 5129 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
4 funss 5112 . 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 103   = wceq 1316  wss 3041  {copab 3958  Fun wfun 5087
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-fun 5095
This theorem is referenced by:  funmpt  5131
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