Theorem fvun 7010

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 6623	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))
2		funfv 7007	. . 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 6181	. . . 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 4944	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∪ ((𝐹 ∪ 𝐺) “ {𝐴}) = ∪ ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})))
7		uniun 4956	. . 3 ⊢ ∪ ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = (∪ (𝐹 “ {𝐴}) ∪ ∪ (𝐺 “ {𝐴}))
8		funfv 7007	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))
9	8	eqcomd 2740	. . . . . 6 ⊢ (Fun 𝐹 → ∪ (𝐹 “ {𝐴}) = (𝐹‘𝐴))
10		funfv 7007	. . . . . . 7 ⊢ (Fun 𝐺 → (𝐺‘𝐴) = ∪ (𝐺 “ {𝐴}))
11	10	eqcomd 2740	. . . . . 6 ⊢ (Fun 𝐺 → ∪ (𝐺 “ {𝐴}) = (𝐺‘𝐴))
12	9, 11	anim12i 612	. . . . 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 4180	. . . 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 2786	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∪ ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))
17	3, 6, 16	3eqtrd 2778	1 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺)‘𝐴) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∪ cun 3968   ∩ cin 3969  ∅c0 4347  {csn 4648  ∪ cuni 4931  dom cdm 5699   “ cima 5702  Fun wfun 6566  ‘cfv 6572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-iota 6524  df-fun 6574  df-fn 6575  df-fv 6580

This theorem is referenced by:  fvun1  7011  undifixp  8988