Theorem unpreima 5643

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 5248	. 2 |- ( Fun F <-> F Fn dom F )
2		elpreima 5637	. . . 4 |- ( F Fn dom F -> ( x e. ( `' F " ( A u. B ) ) <-> ( x e. dom F /\ ( F ` x ) e. ( A u. B ) ) ) )
3		elun 3278	. . . . . . 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 818	. . . . . 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 3278	. . . . . 6 |- ( x e. ( ( `' F " A ) u. ( `' F " B ) ) <-> ( x e. ( `' F " A ) \/ x e. ( `' F " B ) ) )
8		elpreima 5637	. . . . . . 7 |- ( F Fn dom F -> ( x e. ( `' F " A ) <-> ( x e. dom F /\ ( F ` x ) e. A ) ) )
9		elpreima 5637	. . . . . . 7 |- ( F Fn dom F -> ( x e. ( `' F " B ) <-> ( x e. dom F /\ ( F ` x ) e. B ) ) )
10	8, 9	orbi12d 793	. . . . . 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 2175	. 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 708   = wceq 1353   e. wcel 2148   u. cun 3129   `'ccnv 4627   dom cdm 4628   "cima 4631   Fun wfun 5212   Fn wfn 5213   ` cfv 5218

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-fv 5226

This theorem is referenced by: (None)