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Theorem cnvimamptfin 7877
Description: A preimage of a mapping with a finite domain under any class is finite. In contrast to fisuppfi 7893, the range of the mapping needs not to be known. (Contributed by AV, 21-Dec-2018.)
Hypothesis
Ref Expression
cnvimamptfin.n  |-  ( ph  ->  N  e.  Fin )
Assertion
Ref Expression
cnvimamptfin  |-  ( ph  ->  ( `' ( p  e.  N  |->  X )
" Y )  e. 
Fin )
Distinct variable group:    N, p
Allowed substitution hints:    ph( p)    X( p)    Y( p)

Proof of Theorem cnvimamptfin
StepHypRef Expression
1 cnvimamptfin.n . 2  |-  ( ph  ->  N  e.  Fin )
2 cnvimass 5203 . . 3  |-  ( `' ( p  e.  N  |->  X ) " Y
)  C_  dom  ( p  e.  N  |->  X )
3 eqid 2422 . . . 4  |-  ( p  e.  N  |->  X )  =  ( p  e.  N  |->  X )
43dmmptss 5346 . . 3  |-  dom  (
p  e.  N  |->  X )  C_  N
52, 4sstri 3473 . 2  |-  ( `' ( p  e.  N  |->  X ) " Y
)  C_  N
6 ssfi 7794 . 2  |-  ( ( N  e.  Fin  /\  ( `' ( p  e.  N  |->  X ) " Y )  C_  N
)  ->  ( `' ( p  e.  N  |->  X ) " Y
)  e.  Fin )
71, 5, 6sylancl 666 1  |-  ( ph  ->  ( `' ( p  e.  N  |->  X )
" Y )  e. 
Fin )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1868    C_ wss 3436    |-> cmpt 4479   `'ccnv 4848   dom cdm 4849   "cima 4852   Fincfn 7573
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4551  ax-pow 4598  ax-pr 4656  ax-un 6593
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4760  df-id 4764  df-po 4770  df-so 4771  df-fr 4808  df-we 4810  df-xp 4855  df-rel 4856  df-cnv 4857  df-co 4858  df-dm 4859  df-rn 4860  df-res 4861  df-ima 4862  df-ord 5441  df-on 5442  df-lim 5443  df-suc 5444  df-fun 5599  df-fn 5600  df-f 5601  df-f1 5602  df-fo 5603  df-f1o 5604  df-om 6703  df-er 7367  df-en 7574  df-fin 7577
This theorem is referenced by: (None)
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