Theorem fvun 6858

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 6480	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))
2		funfv 6855	. . 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 6054	. . . 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 4853	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∪ ((𝐹 ∪ 𝐺) “ {𝐴}) = ∪ ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})))
7		uniun 4864	. . 3 ⊢ ∪ ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = (∪ (𝐹 “ {𝐴}) ∪ ∪ (𝐺 “ {𝐴}))
8		funfv 6855	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))
9	8	eqcomd 2744	. . . . . 6 ⊢ (Fun 𝐹 → ∪ (𝐹 “ {𝐴}) = (𝐹‘𝐴))
10		funfv 6855	. . . . . . 7 ⊢ (Fun 𝐺 → (𝐺‘𝐴) = ∪ (𝐺 “ {𝐴}))
11	10	eqcomd 2744	. . . . . 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 481	. . . 4 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → (∪ (𝐹 “ {𝐴}) = (𝐹‘𝐴) ∧ ∪ (𝐺 “ {𝐴}) = (𝐺‘𝐴)))
14		uneq12 4092	. . . 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 2790	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∪ ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))
17	3, 6, 16	3eqtrd 2782	1 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺)‘𝐴) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∪ cun 3885   ∩ cin 3886  ∅c0 4256  {csn 4561  ∪ cuni 4839  dom cdm 5589   “ cima 5592  Fun wfun 6427  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441

This theorem is referenced by:  fvun1  6859  undifixp  8722