Theorem fvun 6978

Description: Value of the union of two functions when the domains are separate. (Contributed by FL, 7-Nov-2011.)

Assertion

Ref	Expression
fvun	⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺)‘𝐴) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))

Proof of Theorem fvun

Step	Hyp	Ref	Expression
1		funun 6591	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))
2		funfv 6975	. . 3 ⊢ (Fun (𝐹 ∪ 𝐺) → ((𝐹 ∪ 𝐺)‘𝐴) = ∪ ((𝐹 ∪ 𝐺) “ {𝐴}))
3	1, 2	syl 17	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺)‘𝐴) = ∪ ((𝐹 ∪ 𝐺) “ {𝐴}))
4		imaundir 6150	. . . 4 ⊢ ((𝐹 ∪ 𝐺) “ {𝐴}) = ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴}))
5	4	a1i 11	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺) “ {𝐴}) = ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})))
6	5	unieqd 4900	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∪ ((𝐹 ∪ 𝐺) “ {𝐴}) = ∪ ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})))
7		uniun 4910	. . 3 ⊢ ∪ ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = (∪ (𝐹 “ {𝐴}) ∪ ∪ (𝐺 “ {𝐴}))
8		funfv 6975	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))
9	8	eqcomd 2740	. . . . . 6 ⊢ (Fun 𝐹 → ∪ (𝐹 “ {𝐴}) = (𝐹‘𝐴))
10		funfv 6975	. . . . . . 7 ⊢ (Fun 𝐺 → (𝐺‘𝐴) = ∪ (𝐺 “ {𝐴}))
11	10	eqcomd 2740	. . . . . 6 ⊢ (Fun 𝐺 → ∪ (𝐺 “ {𝐴}) = (𝐺‘𝐴))
12	9, 11	anim12i 613	. . . . 5 ⊢ ((Fun 𝐹 ∧ Fun 𝐺) → (∪ (𝐹 “ {𝐴}) = (𝐹‘𝐴) ∧ ∪ (𝐺 “ {𝐴}) = (𝐺‘𝐴)))
13	12	adantr 480	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → (∪ (𝐹 “ {𝐴}) = (𝐹‘𝐴) ∧ ∪ (𝐺 “ {𝐴}) = (𝐺‘𝐴)))
14		uneq12 4143	. . . 4 ⊢ ((∪ (𝐹 “ {𝐴}) = (𝐹‘𝐴) ∧ ∪ (𝐺 “ {𝐴}) = (𝐺‘𝐴)) → (∪ (𝐹 “ {𝐴}) ∪ ∪ (𝐺 “ {𝐴})) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))
15	13, 14	syl 17	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → (∪ (𝐹 “ {𝐴}) ∪ ∪ (𝐺 “ {𝐴})) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))
16	7, 15	eqtrid 2781	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∪ ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))
17	3, 6, 16	3eqtrd 2773	1 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺)‘𝐴) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∪ cun 3929   ∩ cin 3930  ∅c0 4313  {csn 4606  ∪ cuni 4887  dom cdm 5665   “ cima 5668  Fun wfun 6534  ‘cfv 6540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-id 5558  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6493  df-fun 6542  df-fn 6543  df-fv 6548

This theorem is referenced by:  fvun1  6979  undifixp  8955