Theorem valfunun 14460

Description: Value of the union of two functions when the domains are separate.

Assertion

Ref	Expression
valfunun	|- (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) -> ((F u. G)` A) = ((F` A) u. (G` A)))

Proof of Theorem valfunun

Step	Hyp	Ref	Expression
1		funun 4462	. . 3 |- (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) -> Fun (F u. G))
2		funfv 4731	. . 3 |- (Fun (F u. G) -> ((F u. G)` A) = U.((F u. G)"{A}))
3	1, 2	syl 12	. 2 |- (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) -> ((F u. G)` A) = U.((F u. G)"{A}))
4		imaundir 4329	. . . 4 |- ((F u. G)"{A}) = ((F"{A}) u. (G"{A}))
5	4	a1i 8	. . 3 |- (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) -> ((F u. G)"{A}) = ((F"{A}) u. (G"{A})))
6	5	unieqd 3188	. 2 |- (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) -> U.((F u. G)"{A}) = U.((F"{A}) u. (G"{A})))
7		funfv 4731	. . . . . . 7 |- (Fun F -> (F` A) = U.(F"{A}))
8	7	eqcomd 1889	. . . . . 6 |- (Fun F -> U.(F"{A}) = (F` A))
9		funfv 4731	. . . . . . 7 |- (Fun G -> (G` A) = U.(G"{A}))
10	9	eqcomd 1889	. . . . . 6 |- (Fun G -> U.(G"{A}) = (G` A))
11	8, 10	anim12i 360	. . . . 5 |- ((Fun F /\ Fun G) -> (U.(F"{A}) = (F` A) /\ U.(G"{A}) = (G` A)))
12	11	adantr 425	. . . 4 |- (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) -> (U.(F"{A}) = (F` A) /\ U.(G"{A}) = (G` A)))
13		uneq12 2750	. . . 4 |- ((U.(F"{A}) = (F` A) /\ U.(G"{A}) = (G` A)) -> (U.(F"{A}) u. U.(G"{A})) = ((F` A) u. (G` A)))
14	12, 13	syl 12	. . 3 |- (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) -> (U.(F"{A}) u. U.(G"{A})) = ((F` A) u. (G` A)))
15		uniun 3196	. . 3 |- U.((F"{A}) u. (G"{A})) = (U.(F"{A}) u. U.(G"{A}))
16	14, 15	syl5eq 1940	. 2 |- (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) -> U.((F"{A}) u. (G"{A})) = ((F` A) u. (G` A)))
17	3, 6, 16	3eqtrd 1929	1 |- (((Fun F /\ Fun G) /\ (dom F i^i dom G) = (/)) -> ((F u. G)` A) = ((F` A) u. (G` A)))

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   u. cun 2591   i^i cin 2592  (/)c0 2875  {csn 3044  U.cuni 3177  dom cdm 3986  "cima 3989  Fun wfun 3992  ` cfv 3998

This theorem is referenced by:  unprj 14511

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014

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