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Metamath Proof Explorer

Theorem unpreima

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

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

Proof

Step	Hyp	Ref	Expression
1		funfn	\|- ( Fun F <-> F Fn dom F )
2		elpreima	\|- ( F Fn dom F -> ( x e. ( `' F " ( A u. B ) ) <-> ( x e. dom F /\ ( F ` x ) e. ( A u. B ) ) ) )
3		elun	\|- ( ( F ` x ) e. ( A u. B ) <-> ( ( F ` x ) e. A \/ ( F ` x ) e. B ) )
4	3	anbi2i	\|- ( ( x e. dom F /\ ( F ` x ) e. ( A u. B ) ) <-> ( x e. dom F /\ ( ( F ` x ) e. A \/ ( F ` x ) e. B ) ) )
5		andi	\|- ( ( 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	\|- ( ( 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	\|- ( x e. ( ( `' F " A ) u. ( `' F " B ) ) <-> ( x e. ( `' F " A ) \/ x e. ( `' F " B ) ) )
8		elpreima	\|- ( F Fn dom F -> ( x e. ( `' F " A ) <-> ( x e. dom F /\ ( F ` x ) e. A ) ) )
9		elpreima	\|- ( F Fn dom F -> ( x e. ( `' F " B ) <-> ( x e. dom F /\ ( F ` x ) e. B ) ) )
10	8 9	orbi12d	\|- ( 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	\|- ( 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	\|- ( 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	\|- ( F Fn dom F -> ( x e. ( `' F " ( A u. B ) ) <-> x e. ( ( `' F " A ) u. ( `' F " B ) ) ) )
14	13	eqrdv	\|- ( F Fn dom F -> ( `' F " ( A u. B ) ) = ( ( `' F " A ) u. ( `' F " B ) ) )
15	1 14	sylbi	\|- ( Fun F -> ( `' F " ( A u. B ) ) = ( ( `' F " A ) u. ( `' F " B ) ) )