Theorem funopabeq 6616

Description: A class of ordered pairs of values is a function. (Contributed by NM, 14-Nov-1995.)

Assertion

Ref	Expression
funopabeq	⊢ Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem funopabeq

Step	Hyp	Ref	Expression
1		funopab 6615	. 2 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} ↔ ∀𝑥∃*𝑦 𝑦 = 𝐴)
2		moeq 3729	. 2 ⊢ ∃*𝑦 𝑦 = 𝐴
3	1, 2	mpgbir 1797	1 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}

Syntax hints:   = wceq 1537  ∃*wmo 2541  {copab 5228  Fun wfun 6569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-fun 6577

This theorem is referenced by:  funopab4  6617