Theorem funopabeq 6468

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 6467	. 2 ⊢ (Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} ↔ ∀𝑥∃*𝑦 𝑦 = 𝐴)
2		moeq 3646	. 2 ⊢ ∃*𝑦 𝑦 = 𝐴
3	1, 2	mpgbir 1806	1 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}

Syntax hints:   = wceq 1542  ∃*wmo 2540  {copab 5141  Fun wfun 6426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ral 3071  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-opab 5142  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-fun 6434

This theorem is referenced by:  funopab4  6469