Theorem unpreima 5538

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 5148	. 2 |- ( Fun F <-> F Fn dom F )
2		elpreima 5532	. . . 4 |- ( F Fn dom F -> ( x e. ( `' F " ( A u. B ) ) <-> ( x e. dom F /\ ( F ` x ) e. ( A u. B ) ) ) )
3		elun 3212	. . . . . 6 |- ( x e. ( ( `' F " A ) u. ( `' F " B ) ) <-> ( x e. ( `' F " A ) \/ x e. ( `' F " B ) ) )
4		elpreima 5532	. . . . . . 7 |- ( F Fn dom F -> ( x e. ( `' F " A ) <-> ( x e. dom F /\ ( F ` x ) e. A ) ) )
5		elpreima 5532	. . . . . . 7 |- ( F Fn dom F -> ( x e. ( `' F " B ) <-> ( x e. dom F /\ ( F ` x ) e. B ) ) )
6	4, 5	orbi12d 782	. . . . . 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 ) ) ) )
7	3, 6	syl5bb 191	. . . . 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 ) ) ) )
8		elun 3212	. . . . . . 7 |- ( ( F ` x ) e. ( A u. B ) <-> ( ( F ` x ) e. A \/ ( F ` x ) e. B ) )
9	8	anbi2i 452	. . . . . 6 |- ( ( x e. dom F /\ ( F ` x ) e. ( A u. B ) ) <-> ( x e. dom F /\ ( ( F ` x ) e. A \/ ( F ` x ) e. B ) ) )
10		andi 807	. . . . . 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 ) ) )
11	9, 10	bitri 183	. . . . 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 ) ) )
12	7, 11	syl6rbbr 198	. . . 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 187	. . 3 |- ( F Fn dom F -> ( x e. ( `' F " ( A u. B ) ) <-> x e. ( ( `' F " A ) u. ( `' F " B ) ) ) )
14	13	eqrdv 2135	. 2 |- ( F Fn dom F -> ( `' F " ( A u. B ) ) = ( ( `' F " A ) u. ( `' F " B ) ) )
15	1, 14	sylbi 120	1 |- ( Fun F -> ( `' F " ( A u. B ) ) = ( ( `' F " A ) u. ( `' F " B ) ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 697   = wceq 1331   e. wcel 1480   u. cun 3064   `'ccnv 4533   dom cdm 4534   "cima 4537   Fun wfun 5112   Fn wfn 5113   ` cfv 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-fv 5126

This theorem is referenced by: (None)