Theorem fvun 6924

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 6538	. . 3 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → Fun (𝐹 ∪ 𝐺))
2		funfv 6921	. . 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 6108	. . . 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 4858	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∪ ((𝐹 ∪ 𝐺) “ {𝐴}) = ∪ ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})))
7		uniun 4868	. . 3 ⊢ ∪ ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = (∪ (𝐹 “ {𝐴}) ∪ ∪ (𝐺 “ {𝐴}))
8		funfv 6921	. . . . . . 7 ⊢ (Fun 𝐹 → (𝐹‘𝐴) = ∪ (𝐹 “ {𝐴}))
9	8	eqcomd 2746	. . . . . 6 ⊢ (Fun 𝐹 → ∪ (𝐹 “ {𝐴}) = (𝐹‘𝐴))
10		funfv 6921	. . . . . . 7 ⊢ (Fun 𝐺 → (𝐺‘𝐴) = ∪ (𝐺 “ {𝐴}))
11	10	eqcomd 2746	. . . . . 6 ⊢ (Fun 𝐺 → ∪ (𝐺 “ {𝐴}) = (𝐺‘𝐴))
12	9, 11	anim12i 619	. . . . 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 4100	. . . 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 2787	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ∪ ((𝐹 “ {𝐴}) ∪ (𝐺 “ {𝐴})) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))
17	3, 6, 16	3eqtrd 2779	1 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (dom 𝐹 ∩ dom 𝐺) = ∅) → ((𝐹 ∪ 𝐺)‘𝐴) = ((𝐹‘𝐴) ∪ (𝐺‘𝐴)))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547   ∪ cun 3888   ∩ cin 3889  ∅c0 4268  {csn 4562  ∪ cuni 4845  dom cdm 5625   “ cima 5628  Fun wfun 6486  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fun 6494  df-fn 6495  df-fv 6500

This theorem is referenced by:  fvun1  6925  undifixp  8879