Theorem funrnfi 6919

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 4622	. 2 |- ran A = dom `' A
2		relcnvfi 6918	. . . 4 |- ( ( Rel A /\ A e. Fin ) -> `' A e. Fin )
3	2	3adant2 1011	. . 3 |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> `' A e. Fin )
4		simp2 993	. . 3 |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> Fun `' A )
5		fundmfi 6915	. . 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 2257	1 |- ( ( Rel A /\ Fun `' A /\ A e. Fin ) -> ran A e. Fin )

Syntax hints:   -> wi 4   /\ w3a 973   e. wcel 2141   `'ccnv 4610   dom cdm 4611   ran crn 4612   Rel wrel 4616   Fun wfun 5192   Fincfn 6718

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1st 6119  df-2nd 6120  df-er 6513  df-en 6719  df-fin 6721

This theorem is referenced by:  f1dmvrnfibi  6921