Theorem reldmmpo 6190

Description: The domain of an operation defined by maps-to notation is a relation. (Contributed by Stefan O'Rear, 27-Nov-2014.)

Hypothesis

Ref	Expression
rngop.1	|- F = ( x e. A , y e. B |-> C )

Assertion

Ref	Expression
reldmmpo	|- Rel dom F

Distinct variable groups:   y, A   x, y

Allowed substitution hints:   A( x)   B( x, y)   C( x, y)   F( x, y)

Proof of Theorem reldmmpo

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reldmoprab 6163	. 2 |- Rel dom { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
2		rngop.1	. . . . 5 |- F = ( x e. A , y e. B |-> C )
3		df-mpo 6080	. . . . 5 |- ( x e. A , y e. B |-> C ) = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
4	2, 3	eqtri 2259	. . . 4 |- F = { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
5	4	dmeqi 4977	. . 3 |- dom F = dom { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) }
6	5	releqi 4853	. 2 |- ( Rel dom F <-> Rel dom { <. <. x , y >. , z >. | ( ( x e. A /\ y e. B ) /\ z = C ) } )
7	1, 6	mpbir 146	1 |- Rel dom F

Syntax hints:   /\ wa 104   = wceq 1402   e. wcel 2209   dom cdm 4769   Rel wrel 4774   {coprab 6076   e. cmpo 6077

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-dm 4779  df-oprab 6079  df-mpo 6080

This theorem is referenced by:  reldmmap  6921  reldmsets  13359  reldmress  13394  reldmghm  14022  reldmprds  14148  reldmpsr  14972  reldmmpl  15003