Theorem relcnvfi 7139

Description: If a relation is finite, its converse is as well. (Contributed by Jim Kingdon, 5-Feb-2022.)

Assertion

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

Proof of Theorem relcnvfi

Step	Hyp	Ref	Expression
1		dfrel2 5187	. . . . 5 |- ( Rel A <-> `' `' A = A )
2	1	biimpi 120	. . . 4 |- ( Rel A -> `' `' A = A )
3	2	adantr 276	. . 3 |- ( ( Rel A /\ A e. Fin ) -> `' `' A = A )
4		simpr 110	. . 3 |- ( ( Rel A /\ A e. Fin ) -> A e. Fin )
5	3, 4	eqeltrd 2308	. 2 |- ( ( Rel A /\ A e. Fin ) -> `' `' A e. Fin )
6		relcnv 5114	. . . 4 |- Rel `' A
7		cnvexg 5274	. . . 4 |- ( A e. Fin -> `' A e. _V )
8		cnven 6982	. . . 4 |- ( ( Rel `' A /\ `' A e. _V ) -> `' A ~~ `' `' A )
9	6, 7, 8	sylancr 414	. . 3 |- ( A e. Fin -> `' A ~~ `' `' A )
10	9	adantl 277	. 2 |- ( ( Rel A /\ A e. Fin ) -> `' A ~~ `' `' A )
11		enfii 7060	. 2 |- ( ( `' `' A e. Fin /\ `' A ~~ `' `' A ) -> `' A e. Fin )
12	5, 10, 11	syl2anc 411	1 |- ( ( Rel A /\ A e. Fin ) -> `' A e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   _Vcvv 2802   class class class wbr 4088   `'ccnv 4724   Rel wrel 4730   ~~ cen 6906   Fincfn 6908

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1st 6302  df-2nd 6303  df-er 6701  df-en 6909  df-fin 6911

This theorem is referenced by:  funrnfi  7140  fsumcnv  11997  fprodcnv  12185