Theorem reldmdsmm 19373

Description: The direct sum is a well-behaved binary operator. (Contributed by Stefan O'Rear, 7-Jan-2015.)

Assertion

Ref	Expression
reldmdsmm	|- Rel dom (+)m

Proof of Theorem reldmdsmm

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

Step	Hyp	Ref	Expression
1		df-dsmm 19372	. 2 |- (+)m = ( s e. _V , r e. _V |-> ( ( s X_s r ) ↾s { f e. X_ x e. dom r ( Base ` ( r ` x ) ) | { x e. dom r | ( f ` x ) =/= ( 0g ` ( r ` x ) ) } e. Fin } ) )
2	1	reldmmpt2 6426	1 |- Rel dom (+)m

Syntax hints:   e. wcel 1904   =/= wne 2641   {crab 2760   _Vcvv 3031   dom cdm 4839   Rel wrel 4844   ` cfv 5589  (class class class)co 6308   X_cixp 7540   Fincfn 7587   Basecbs 15199   ↾_s cress 15200   0gc0g 15416   X__scprds 15422   (+)_m cdsmm 19371

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-dm 4849  df-oprab 6312  df-mpt2 6313  df-dsmm 19372

This theorem is referenced by:  dsmmval  19374  dsmmval2  19376