Metamath Proof Explorer

Theorem uniimadom

Description: An upper bound for the cardinality of the union of an image. Theorem 10.48 of TakeutiZaring p. 99. (Contributed by NM, 25-Mar-2006)

Ref Expression
Hypotheses uniimadom.1 
|- A e. _V
uniimadom.2 
|- B e. _V
Assertion uniimadom 
|- ( ( Fun F /\ A. x e. A ( F ` x ) ~<_ B ) -> U. ( F " A ) ~<_ ( A X. B ) )

Proof

Step Hyp Ref Expression
1 uniimadom.1 
 |-  A e. _V
2 uniimadom.2 
 |-  B e. _V
3 1 funimaex 
 |-  ( Fun F -> ( F " A ) e. _V )
4 3 adantr 
 |-  ( ( Fun F /\ A. x e. A ( F ` x ) ~<_ B ) -> ( F " A ) e. _V )
5 fvelima 
 |-  ( ( Fun F /\ y e. ( F " A ) ) -> E. x e. A ( F ` x ) = y )
6 5 ex 
 |-  ( Fun F -> ( y e. ( F " A ) -> E. x e. A ( F ` x ) = y ) )
7 breq1 
 |-  ( ( F ` x ) = y -> ( ( F ` x ) ~<_ B <-> y ~<_ B ) )
8 7 biimpd 
 |-  ( ( F ` x ) = y -> ( ( F ` x ) ~<_ B -> y ~<_ B ) )
9 8 reximi 
 |-  ( E. x e. A ( F ` x ) = y -> E. x e. A ( ( F ` x ) ~<_ B -> y ~<_ B ) )
10 r19.36v 
 |-  ( E. x e. A ( ( F ` x ) ~<_ B -> y ~<_ B ) -> ( A. x e. A ( F ` x ) ~<_ B -> y ~<_ B ) )
11 9 10 syl 
 |-  ( E. x e. A ( F ` x ) = y -> ( A. x e. A ( F ` x ) ~<_ B -> y ~<_ B ) )
12 6 11 syl6 
 |-  ( Fun F -> ( y e. ( F " A ) -> ( A. x e. A ( F ` x ) ~<_ B -> y ~<_ B ) ) )
13 12 com23 
 |-  ( Fun F -> ( A. x e. A ( F ` x ) ~<_ B -> ( y e. ( F " A ) -> y ~<_ B ) ) )
14 13 imp 
 |-  ( ( Fun F /\ A. x e. A ( F ` x ) ~<_ B ) -> ( y e. ( F " A ) -> y ~<_ B ) )
15 14 ralrimiv 
 |-  ( ( Fun F /\ A. x e. A ( F ` x ) ~<_ B ) -> A. y e. ( F " A ) y ~<_ B )
16 unidom 
 |-  ( ( ( F " A ) e. _V /\ A. y e. ( F " A ) y ~<_ B ) -> U. ( F " A ) ~<_ ( ( F " A ) X. B ) )
17 4 15 16 syl2anc 
 |-  ( ( Fun F /\ A. x e. A ( F ` x ) ~<_ B ) -> U. ( F " A ) ~<_ ( ( F " A ) X. B ) )
18 imadomg 
 |-  ( A e. _V -> ( Fun F -> ( F " A ) ~<_ A ) )
19 1 18 ax-mp 
 |-  ( Fun F -> ( F " A ) ~<_ A )
20 2 xpdom1 
 |-  ( ( F " A ) ~<_ A -> ( ( F " A ) X. B ) ~<_ ( A X. B ) )
21 19 20 syl 
 |-  ( Fun F -> ( ( F " A ) X. B ) ~<_ ( A X. B ) )
22 21 adantr 
 |-  ( ( Fun F /\ A. x e. A ( F ` x ) ~<_ B ) -> ( ( F " A ) X. B ) ~<_ ( A X. B ) )
23 domtr 
 |-  ( ( U. ( F " A ) ~<_ ( ( F " A ) X. B ) /\ ( ( F " A ) X. B ) ~<_ ( A X. B ) ) -> U. ( F " A ) ~<_ ( A X. B ) )
24 17 22 23 syl2anc 
 |-  ( ( Fun F /\ A. x e. A ( F ` x ) ~<_ B ) -> U. ( F " A ) ~<_ ( A X. B ) )