Theorem funopab4 5389

Description: A class of ordered pairs of values in the form used by df-mpt 4173 is a function. (Contributed by NM, 17-Feb-2013.)

Assertion

Ref	Expression
funopab4	⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)}

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem funopab4

Step	Hyp	Ref	Expression
1		simpr 110	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
2	1	ssopab2i 4396	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
3		funopabeq 5388	. 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
4		funss 5371	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} → (Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴} → Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)}))
5	2, 3, 4	mp2 16	1 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)}

Syntax hints:   ∧ wa 104   = wceq 1398   ⊆ wss 3211  {copab 4170  Fun wfun 5346

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-id 4414  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-fun 5354

This theorem is referenced by:  funmpt  5390