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Theorem funopab4 5839
 Description: A class of ordered pairs of values in the form used by df-mpt 4645 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 476 . . 3 ((𝜑𝑦 = 𝐴) → 𝑦 = 𝐴)
21ssopab2i 4928 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
3 funopabeq 5838 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
4 funss 5822 . 2 ({⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} → (Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} → Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)}))
52, 3, 4mp2 9 1 Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475   ⊆ wss 3540  {copab 4642  Fun wfun 5798 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-fun 5806 This theorem is referenced by:  funmpt  5840  hartogslem1  8330
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