Theorem brstruct 13339

Description: The structure relation is a relation. (Contributed by Mario Carneiro, 29-Aug-2015.)

Assertion

Ref	Expression
brstruct	|- Rel Struct

Proof of Theorem brstruct

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

Step	Hyp	Ref	Expression
1		df-struct 13332	. 2 |- Struct = { <. f , x >. | ( x e. ( <_ i^i ( NN X. NN ) ) /\ Fun ( f \ { (/) } ) /\ dom f C_ ( ... ` x ) ) }
2	1	relopabi 4900	1 |- Rel Struct

Syntax hints:   /\ w3a 1009   e. wcel 2209   \ cdif 3217   i^i cin 3219   C_ wss 3220   (/)c0 3520   {csn 3705   X. cxp 4767   dom cdm 4769   Rel wrel 4774   Fun wfun 5366   ` cfv 5372   <_ cle 8351   NNcn 9283   ...cfz 10390   Struct cstr 13326

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-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-opab 4188  df-xp 4775  df-rel 4776  df-struct 13332

This theorem is referenced by:  isstruct2im  13340  structex  13342