Theorem reldmdsmm 18891

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 18890	. 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 6412	1 |- Rel dom (+)m

Syntax hints:   e. wcel 1819   =/= wne 2652   {crab 2811   _Vcvv 3109   dom cdm 5008   Rel wrel 5013   ` cfv 5594  (class class class)co 6296   X_cixp 7488   Fincfn 7535   Basecbs 14644   ↾_s cress 14645   0gc0g 14857   X__scprds 14863   (+)_m cdsmm 18889

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-dm 5018  df-oprab 6300  df-mpt2 6301  df-dsmm 18890

This theorem is referenced by:  dsmmval  18892  dsmmval2  18894