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Theorem funopab4 5923
Description: A class of ordered pairs of values in the form used by df-mpt 4728 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 477 . . 3 ((𝜑𝑦 = 𝐴) → 𝑦 = 𝐴)
21ssopab2i 5001 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
3 funopabeq 5922 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
4 funss 5905 . 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 384   = wceq 1482  wss 3572  {copab 4710  Fun wfun 5880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-fun 5888
This theorem is referenced by:  funmpt  5924  hartogslem1  8444
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