Theorem rnghmrcl 41889

Description: Reverse closure of a non-unital ring homomorphism. (Contributed by AV, 22-Feb-2020.)

Assertion

Ref	Expression
rnghmrcl	|- ( F e. ( R RngHomo S ) -> ( R e. Rng /\ S e. Rng ) )

Proof of Theorem rnghmrcl

Dummy variables x s r v w f y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rnghomo 41887	. 2 |- RngHomo = ( r e. Rng , s e. Rng |-> [_ ( Base ` r ) / v ]_ [_ ( Base ` s ) / w ]_ { f e. ( w ^m v ) | A. x e. v A. y e. v ( ( f ` ( x ( +g ` r ) y ) ) = ( ( f ` x ) ( +g ` s ) ( f ` y ) ) /\ ( f ` ( x ( .r ` r ) y ) ) = ( ( f ` x ) ( .r ` s ) ( f ` y ) ) ) } )
2	1	elmpt2cl 6876	1 |- ( F e. ( R RngHomo S ) -> ( R e. Rng /\ S e. Rng ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   [_csb 3533   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   Basecbs 15857   +g cplusg 15941   .rcmulr 15942  Rngcrng 41874   RngHomo crngh 41885

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-ne 2795  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-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-rnghomo 41887

This theorem is referenced by:  isrnghm  41892  rnghmf1o  41903  rnghmco  41907