Theorem fununfun 6588

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 6557	. . 3 ⊢ (Fun (𝐹 ∪ 𝐺) → Rel (𝐹 ∪ 𝐺))
2		relun 5800	. . 3 ⊢ (Rel (𝐹 ∪ 𝐺) ↔ (Rel 𝐹 ∧ Rel 𝐺))
3	1, 2	sylib 221	. 2 ⊢ (Fun (𝐹 ∪ 𝐺) → (Rel 𝐹 ∧ Rel 𝐺))
4		simpl 488	. . . . 5 ⊢ ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐹)
5		fununmo 6587	. . . . . 6 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐹𝑦)
6	5	alrimiv 1960	. . . . 5 ⊢ (Fun (𝐹 ∪ 𝐺) → ∀𝑥∃*𝑦 𝑥𝐹𝑦)
7	4, 6	anim12i 625	. . . 4 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
8		dffun6 6551	. . . 4 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∃*𝑦 𝑥𝐹𝑦))
9	7, 8	sylibr 237	. . 3 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → Fun 𝐹)
10		simpr 490	. . . . 5 ⊢ ((Rel 𝐹 ∧ Rel 𝐺) → Rel 𝐺)
11		uncom 4112	. . . . . . . 8 ⊢ (𝐹 ∪ 𝐺) = (𝐺 ∪ 𝐹)
12	11	funeqi 6561	. . . . . . 7 ⊢ (Fun (𝐹 ∪ 𝐺) ↔ Fun (𝐺 ∪ 𝐹))
13		fununmo 6587	. . . . . . 7 ⊢ (Fun (𝐺 ∪ 𝐹) → ∃*𝑦 𝑥𝐺𝑦)
14	12, 13	sylbi 220	. . . . . 6 ⊢ (Fun (𝐹 ∪ 𝐺) → ∃*𝑦 𝑥𝐺𝑦)
15	14	alrimiv 1960	. . . . 5 ⊢ (Fun (𝐹 ∪ 𝐺) → ∀𝑥∃*𝑦 𝑥𝐺𝑦)
16	10, 15	anim12i 625	. . . 4 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦))
17		dffun6 6551	. . . 4 ⊢ (Fun 𝐺 ↔ (Rel 𝐺 ∧ ∀𝑥∃*𝑦 𝑥𝐺𝑦))
18	16, 17	sylibr 237	. . 3 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → Fun 𝐺)
19	9, 18	jca 521	. 2 ⊢ (((Rel 𝐹 ∧ Rel 𝐺) ∧ Fun (𝐹 ∪ 𝐺)) → (Fun 𝐹 ∧ Fun 𝐺))
20	3, 19	mpancom 701	1 ⊢ (Fun (𝐹 ∪ 𝐺) → (Fun 𝐹 ∧ Fun 𝐺))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568  ∃*wmo 2567   ∪ cun 3904   class class class wbr 5111  Rel wrel 5668  Fun wfun 6534

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 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

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-sb 2100  df-mo 2569  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-fun 6542

This theorem is used by:  fsuppunbi  9356