Theorem fununfun 6616

Description: If the union of classes is a function, the classes itselves are functions. (Contributed by AV, 18-Jul-2019.)

Assertion

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

Proof of Theorem fununfun

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		funrel 6585	. . 3 ⊢ (Fun (𝐹 ∪ 𝐺) → Rel (𝐹 ∪ 𝐺))
2		relun 5824	. . 3 ⊢ (Rel (𝐹 ∪ 𝐺) ↔ (Rel 𝐹 ∧ Rel 𝐺))
3	1, 2	sylib 218	. 2 ⊢ (Fun (𝐹 ∪ 𝐺) → (Rel 𝐹 ∧ Rel 𝐺))
4		simpl 482	. . . . 5 ⊢ ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐹)
5		fununmo 6615	. . . . . 6 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐹𝑦)
6	5	alrimiv 1925	. . . . 5 ⊢ (Fun (𝐹 ∪ 𝐺) → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
7	4, 6	anim12i 613	. . . 4 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
8		dffun6 6576	. . . 4 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
9	7, 8	sylibr 234	. . 3 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → Fun 𝐹)
10		simpr 484	. . . . 5 ⊢ ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐺)
11		uncom 4168	. . . . . . . 8 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹)
12	11	funeqi 6589	. . . . . . 7 ⊢ (Fun (𝐹 ∪ 𝐺) ↔ Fun (𝐺 ∪ 𝐹))
13		fununmo 6615	. . . . . . 7 ⊢ (Fun (𝐺 ∪ 𝐹) → ∃*𝑦 𝑥𝐺𝑦)
14	12, 13	sylbi 217	. . . . . 6 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐺𝑦)
15	14	alrimiv 1925	. . . . 5 ⊢ (Fun (𝐹 ∪ 𝐺) → ∀𝑥∃*𝑦 𝑥𝐺𝑦)
16	10, 15	anim12i 613	. . . 4 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦))
17		dffun6 6576	. . . 4 ⊢ (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦))
18	16, 17	sylibr 234	. . 3 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → Fun 𝐺)
19	9, 18	jca 511	. 2 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Fun 𝐹 ∧ Fun 𝐺))
20	3, 19	mpancom 688	1 ⊢ (Fun (𝐹 ∪ 𝐺) → (Fun 𝐹 ∧ Fun 𝐺))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1535  ∃*wmo 2536   ∪ cun 3961   class class class wbr 5148  Rel wrel 5694  Fun wfun 6557

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-fun 6565

This theorem is referenced by:  fsuppunbi  9427