Theorem unpreima 5807

Description: Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
unpreima	|- ( Fun F -> ( `' F " ( A u. B ) ) = ( ( `' F " A ) u. ( `' F " B ) ) )

Proof of Theorem unpreima

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funfn 5387	. 2 |- ( Fun F <-> F Fn dom F )
2		elpreima 5802	. . . 4 |- ( F Fn dom F -> ( x e. ( `' F " ( A u. B ) ) <-> ( x e. dom F /\ ( F ` x ) e. ( A u. B ) ) ) )
3		elun 3364	. . . . . . 7 |- ( ( F ` x ) e. ( A u. B ) <-> ( ( F ` x ) e. A \/ ( F ` x ) e. B ) )
4	3	anbi2i 457	. . . . . 6 |- ( ( x e. dom F /\ ( F ` x ) e. ( A u. B ) ) <-> ( x e. dom F /\ ( ( F ` x ) e. A \/ ( F ` x ) e. B ) ) )
5		andi 826	. . . . . 6 |- ( ( x e. dom F /\ ( ( F ` x ) e. A \/ ( F ` x ) e. B ) ) <-> ( ( x e. dom F /\ ( F ` x ) e. A ) \/ ( x e. dom F /\ ( F ` x ) e. B ) ) )
6	4, 5	bitri 184	. . . . 5 |- ( ( x e. dom F /\ ( F ` x ) e. ( A u. B ) ) <-> ( ( x e. dom F /\ ( F ` x ) e. A ) \/ ( x e. dom F /\ ( F ` x ) e. B ) ) )
7		elun 3364	. . . . . 6 |- ( x e. ( ( `' F " A ) u. ( `' F " B ) ) <-> ( x e. ( `' F " A ) \/ x e. ( `' F " B ) ) )
8		elpreima 5802	. . . . . . 7 |- ( F Fn dom F -> ( x e. ( `' F " A ) <-> ( x e. dom F /\ ( F ` x ) e. A ) ) )
9		elpreima 5802	. . . . . . 7 |- ( F Fn dom F -> ( x e. ( `' F " B ) <-> ( x e. dom F /\ ( F ` x ) e. B ) ) )
10	8, 9	orbi12d 801	. . . . . 6 |- ( F Fn dom F -> ( ( x e. ( `' F " A ) \/ x e. ( `' F " B ) ) <-> ( ( x e. dom F /\ ( F ` x ) e. A ) \/ ( x e. dom F /\ ( F ` x ) e. B ) ) ) )
11	7, 10	bitrid 192	. . . . 5 |- ( F Fn dom F -> ( x e. ( ( `' F " A ) u. ( `' F " B ) ) <-> ( ( x e. dom F /\ ( F ` x ) e. A ) \/ ( x e. dom F /\ ( F ` x ) e. B ) ) ) )
12	6, 11	bitr4id 199	. . . 4 |- ( F Fn dom F -> ( ( x e. dom F /\ ( F ` x ) e. ( A u. B ) ) <-> x e. ( ( `' F " A ) u. ( `' F " B ) ) ) )
13	2, 12	bitrd 188	. . 3 |- ( F Fn dom F -> ( x e. ( `' F " ( A u. B ) ) <-> x e. ( ( `' F " A ) u. ( `' F " B ) ) ) )
14	13	eqrdv 2232	. 2 |- ( F Fn dom F -> ( `' F " ( A u. B ) ) = ( ( `' F " A ) u. ( `' F " B ) ) )
15	1, 14	sylbi 121	1 |- ( Fun F -> ( `' F " ( A u. B ) ) = ( ( `' F " A ) u. ( `' F " B ) ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2205   u. cun 3212   `'ccnv 4753   dom cdm 4754   "cima 4757   Fun wfun 5351   Fn wfn 5352   ` cfv 5357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-fv 5365

This theorem is referenced by: (None)