Theorem funopab4 6577

Description: A class of ordered pairs of values in the form used by df-mpt 5187 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 490	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝐴) → 𝑦 = 𝐴)
2	1	ssopab2i 5525	. 2 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝑦 = 𝐴)} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
3		funopabeq 6576	. 2 ⊢ Fun {⟨𝑥, 𝑦⟩ ∣ 𝑦 = 𝐴}
4		funss 6558	. 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)}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570   ⊆ wss 3899  {copab 5167  Fun wfun 6532

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-fun 6540

This theorem is used by:  funmpt  6578  hartogslem1  9536