Theorem ovrcl 6686

Description: Reverse closure for an operation value. (Contributed by Mario Carneiro, 5-May-2015.)

Hypothesis

Ref	Expression
ovprc1.1	|- Rel dom F

Assertion

Ref	Expression
ovrcl	|- ( C e. ( A F B ) -> ( A e. _V /\ B e. _V ) )

Proof of Theorem ovrcl

Step	Hyp	Ref	Expression
1		n0i 3920	. 2 |- ( C e. ( A F B ) -> -. ( A F B ) = (/) )
2		ovprc1.1	. . 3 |- Rel dom F
3	2	ovprc 6683	. 2 |- ( -. ( A e. _V /\ B e. _V ) -> ( A F B ) = (/) )
4	1, 3	nsyl2 142	1 |- ( C e. ( A F B ) -> ( A e. _V /\ B e. _V ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   (/)c0 3915   dom cdm 5114   Rel wrel 5119  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  cda1dif  8998  smatrcl  29862