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Metamath Proof Explorer

Theorem imadomg

Description: An image of a function under a set is dominated by the set. Proposition 10.34 of TakeutiZaring p. 92. (Contributed by NM, 23-Jul-2004)

Ref Expression
Assertion imadomg 
|- ( A e. B -> ( Fun F -> ( F " A ) ~<_ A ) )

Proof

Step Hyp Ref Expression
1 df-ima 
 |-  ( F " A ) = ran ( F |` A )
2 resfunexg 
 |-  ( ( Fun F /\ A e. B ) -> ( F |` A ) e. _V )
3 2 dmexd 
 |-  ( ( Fun F /\ A e. B ) -> dom ( F |` A ) e. _V )
4 funres 
 |-  ( Fun F -> Fun ( F |` A ) )
5 funforn 
 |-  ( Fun ( F |` A ) <-> ( F |` A ) : dom ( F |` A ) -onto-> ran ( F |` A ) )
6 4 5 sylib 
 |-  ( Fun F -> ( F |` A ) : dom ( F |` A ) -onto-> ran ( F |` A ) )
7 6 adantr 
 |-  ( ( Fun F /\ A e. B ) -> ( F |` A ) : dom ( F |` A ) -onto-> ran ( F |` A ) )
8 fodomg 
 |-  ( dom ( F |` A ) e. _V -> ( ( F |` A ) : dom ( F |` A ) -onto-> ran ( F |` A ) -> ran ( F |` A ) ~<_ dom ( F |` A ) ) )
9 3 7 8 sylc 
 |-  ( ( Fun F /\ A e. B ) -> ran ( F |` A ) ~<_ dom ( F |` A ) )
10 1 9 eqbrtrid 
 |-  ( ( Fun F /\ A e. B ) -> ( F " A ) ~<_ dom ( F |` A ) )
11 10 expcom 
 |-  ( A e. B -> ( Fun F -> ( F " A ) ~<_ dom ( F |` A ) ) )
12 dmres 
 |-  dom ( F |` A ) = ( A i^i dom F )
13 inss1 
 |-  ( A i^i dom F ) C_ A
14 12 13 eqsstri 
 |-  dom ( F |` A ) C_ A
15 ssdomg 
 |-  ( A e. B -> ( dom ( F |` A ) C_ A -> dom ( F |` A ) ~<_ A ) )
16 14 15 mpi 
 |-  ( A e. B -> dom ( F |` A ) ~<_ A )
17 domtr 
 |-  ( ( ( F " A ) ~<_ dom ( F |` A ) /\ dom ( F |` A ) ~<_ A ) -> ( F " A ) ~<_ A )
18 16 17 sylan2 
 |-  ( ( ( F " A ) ~<_ dom ( F |` A ) /\ A e. B ) -> ( F " A ) ~<_ A )
19 18 expcom 
 |-  ( A e. B -> ( ( F " A ) ~<_ dom ( F |` A ) -> ( F " A ) ~<_ A ) )
20 11 19 syld 
 |-  ( A e. B -> ( Fun F -> ( F " A ) ~<_ A ) )