Theorem funrnfi 7222

Description: The range of a finite relation is finite if its converse is a function. (Contributed by Jim Kingdon, 5-Feb-2022.)

Assertion

Ref	Expression
funrnfi	|- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> ran A e. Fin )

Proof of Theorem funrnfi

Step	Hyp	Ref	Expression
1		df-rn 4765	. 2 |- ran A = dom `' A
2		relcnvfi 7221	. . . 4 |- ( ( Rel A /\ A e. Fin ) -> `' A e. Fin )
3	2	3adant2 1043	. . 3 |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> `' A e. Fin )
4		simp2 1025	. . 3 |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> Fun `' A )
5		fundmfi 7217	. . 3 |- ( ( `' A e. Fin /\ Fun `' A ) -> dom `' A e. Fin )
6	3, 4, 5	syl2anc 411	. 2 |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> dom `' A e. Fin )
7	1, 6	eqeltrid 2321	1 |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> ran A e. Fin )

Syntax hints:   -> wi 4   /\ w3a 1005   e. wcel 2205   `'ccnv 4753   dom cdm 4754   ran crn 4755   Rel wrel 4759   Fun wfun 5351   Fincfn 6988

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-1st 6347  df-2nd 6348  df-er 6780  df-en 6989  df-fin 6991

This theorem is referenced by:  f1dmvrnfibi  7224  4sqlemffi  13119