Theorem funrnfi 6943

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 4639	. 2 |- ran A = dom `' A
2		relcnvfi 6942	. . . 4 |- ( ( Rel A /\ A e. Fin ) -> `' A e. Fin )
3	2	3adant2 1016	. . 3 |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> `' A e. Fin )
4		simp2 998	. . 3 |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> Fun `' A )
5		fundmfi 6939	. . 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 2264	1 |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> ran A e. Fin )

Syntax hints:   -> wi 4   /\ w3a 978   e. wcel 2148   `'ccnv 4627   dom cdm 4628   ran crn 4629   Rel wrel 4633   Fun wfun 5212   Fincfn 6742

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-sbc 2965  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6143  df-2nd 6144  df-er 6537  df-en 6743  df-fin 6745

This theorem is referenced by:  f1dmvrnfibi  6945