Theorem fununfun 6616

Description: If the union of classes is a function, the classes themselves are functions. (Contributed by AV, 18-Jul-2019.) Shorten proof and remove dependency on ax-sep 5255 and ax-pr 5398. (Revised by Eric Schmidt, 24-Oct-2026.)

Assertion

Ref	Expression
fununfun	⊢ (Fun (𝐹 ∪ 𝐺) → (Fun 𝐹 ∧ Fun 𝐺))

Proof of Theorem fununfun

Step	Hyp	Ref	Expression
1		ssun1 4179	. . 3 ⊢ 𝐹 ⊆ (𝐹 ∪ 𝐺)
2		funss 6586	. . 3 ⊢ (𝐹 ⊆ (𝐹 ∪ 𝐺) → (Fun (𝐹 ∪ 𝐺) → Fun 𝐹))
3	1, 2	ax-mp 5	. 2 ⊢ (Fun (𝐹 ∪ 𝐺) → Fun 𝐹)
4		ssun2 4180	. . 3 ⊢ 𝐺 ⊆ (𝐹 ∪ 𝐺)
5		funss 6586	. . 3 ⊢ (𝐺 ⊆ (𝐹 ∪ 𝐺) → (Fun (𝐹 ∪ 𝐺) → Fun 𝐺))
6	4, 5	ax-mp 5	. 2 ⊢ (Fun (𝐹 ∪ 𝐺) → Fun 𝐺)
7	3, 6	jca 521	1 ⊢ (Fun (𝐹 ∪ 𝐺) → (Fun 𝐹 ∧ Fun 𝐺))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ⊆ wss 3900   ∪ cun 4152  Fun wfun 6560

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-ss 3901  df-un 4154  df-br 5109  df-opab 5174  df-rel 5665  df-cnv 5666  df-co 5667  df-fun 6568

This theorem is used by:  fsuppunbi  9418