Theorem funrnfi 6907

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 4615	. 2 |- ran A = dom `' A
2		relcnvfi 6906	. . . 4 |- ( ( Rel A /\ A e. Fin ) -> `' A e. Fin )
3	2	3adant2 1006	. . 3 |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> `' A e. Fin )
4		simp2 988	. . 3 |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> Fun `' A )
5		fundmfi 6903	. . 3 |- ( ( `' A e. Fin /\ Fun `' A ) -> dom `' A e. Fin )
6	3, 4, 5	syl2anc 409	. 2 |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> dom `' A e. Fin )
7	1, 6	eqeltrid 2253	1 |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> ran A e. Fin )

Syntax hints:   -> wi 4   /\ w3a 968   e. wcel 2136   `'ccnv 4603   dom cdm 4604   ran crn 4605   Rel wrel 4609   Fun wfun 5182   Fincfn 6706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-sbc 2952  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1st 6108  df-2nd 6109  df-er 6501  df-en 6707  df-fin 6709

This theorem is referenced by:  f1dmvrnfibi  6909