Theorem funopab4 6386

Description: A class of ordered pairs of values in the form used by df-mpt 5121 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 488	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
2	1	ssopab2i 5415	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
3		funopabeq 6385	. 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
4		funss 6368	. 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 9	1 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)}

Syntax hints:   ∧ wa 399   = wceq 1542   ⊆ wss 3853  {copab 5102  Fun wfun 6343

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 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-v 3402  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-fun 6351

This theorem is referenced by:  funmpt  6387  hartogslem1  9091