Theorem funopabeq 6586

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 6585	. 2 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} ↔ ∀𝑥∃*𝑦 𝑦 = 𝐴)
2		moeq 3702	. 2 ⊢ ∃*𝑦 𝑦 = 𝐴
3	1, 2	mpgbir 1794	1 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}

Syntax hints:   = wceq 1534  ∃*wmo 2527  {copab 5207  Fun wfun 6539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3466  df-dif 3951  df-un 3953  df-ss 3965  df-nul 4325  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5146  df-opab 5208  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-fun 6547

This theorem is referenced by:  funopab4  6587