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Theorem funopab4 6595
Description: A class of ordered pairs of values in the form used by df-mpt 5236 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 483 . . 3 ((𝜑𝑦 = 𝐴) → 𝑦 = 𝐴)
21ssopab2i 5556 . 2 {⟨𝑥, 𝑦⟩ ∣ (𝜑𝑦 = 𝐴)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
3 funopabeq 6594 . 2 Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
4 funss 6577 . 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 394   = wceq 1533  wss 3949  {copab 5214  Fun wfun 6547
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-fun 6555
This theorem is referenced by:  funmpt  6596  hartogslem1  9573
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