Metamath Proof Explorer

Theorem imadomnum

Description: A version of imadomg that does not require the axiom of choice ax-ac . (Contributed by Vincent Gonzalez, 25-Aug-2026)

Ref Expression
Assertion imadomnum 
|- ( A e. dom card -> ( Fun F -> ( F " A ) ~<_ A ) )

Proof

Step Hyp Ref Expression
1 df-ima 
 |-  ( F " A ) = ran ( F |` A )
2 inss1 
 |-  ( A i^i dom F ) C_ A
3 ssnum 
 |-  ( ( A e. dom card /\ ( A i^i dom F ) C_ A ) -> ( A i^i dom F ) e. dom card )
4 2 3 mpan2 
 |-  ( A e. dom card -> ( A i^i dom F ) e. dom card )
5 4 adantr 
 |-  ( ( A e. dom card /\ Fun F ) -> ( A i^i dom F ) e. dom card )
6 funres 
 |-  ( Fun F -> Fun ( F |` A ) )
7 funfn 
 |-  ( Fun ( F |` A ) <-> ( F |` A ) Fn dom ( F |` A ) )
8 6 7 sylib 
 |-  ( Fun F -> ( F |` A ) Fn dom ( F |` A ) )
9 dmres 
 |-  dom ( F |` A ) = ( A i^i dom F )
10 9 fneq2i 
 |-  ( ( F |` A ) Fn dom ( F |` A ) <-> ( F |` A ) Fn ( A i^i dom F ) )
11 8 10 sylib 
 |-  ( Fun F -> ( F |` A ) Fn ( A i^i dom F ) )
12 11 adantl 
 |-  ( ( A e. dom card /\ Fun F ) -> ( F |` A ) Fn ( A i^i dom F ) )
13 dffn4 
 |-  ( ( F |` A ) Fn ( A i^i dom F ) <-> ( F |` A ) : ( A i^i dom F ) -onto-> ran ( F |` A ) )
14 fodomnum 
 |-  ( ( A i^i dom F ) e. dom card -> ( ( F |` A ) : ( A i^i dom F ) -onto-> ran ( F |` A ) -> ran ( F |` A ) ~<_ ( A i^i dom F ) ) )
15 13 14 biimtrid 
 |-  ( ( A i^i dom F ) e. dom card -> ( ( F |` A ) Fn ( A i^i dom F ) -> ran ( F |` A ) ~<_ ( A i^i dom F ) ) )
16 5 12 15 sylc 
 |-  ( ( A e. dom card /\ Fun F ) -> ran ( F |` A ) ~<_ ( A i^i dom F ) )
17 1 16 eqbrtrid 
 |-  ( ( A e. dom card /\ Fun F ) -> ( F " A ) ~<_ ( A i^i dom F ) )
18 elex 
 |-  ( A e. dom card -> A e. _V )
19 ssdomg 
 |-  ( A e. _V -> ( ( A i^i dom F ) C_ A -> ( A i^i dom F ) ~<_ A ) )
20 18 19 syl 
 |-  ( A e. dom card -> ( ( A i^i dom F ) C_ A -> ( A i^i dom F ) ~<_ A ) )
21 2 20 mpi 
 |-  ( A e. dom card -> ( A i^i dom F ) ~<_ A )
22 21 adantr 
 |-  ( ( A e. dom card /\ Fun F ) -> ( A i^i dom F ) ~<_ A )
23 domtr 
 |-  ( ( ( F " A ) ~<_ ( A i^i dom F ) /\ ( A i^i dom F ) ~<_ A ) -> ( F " A ) ~<_ A )
24 17 22 23 syl2anc 
 |-  ( ( A e. dom card /\ Fun F ) -> ( F " A ) ~<_ A )
25 24 ex 
 |-  ( A e. dom card -> ( Fun F -> ( F " A ) ~<_ A ) )