Theorem reldcat 10666

Description: The domain of a category is a relation.

Assertion

Ref	Expression
reldcat	|- Rel dom Cat

Proof of Theorem reldcat

Step	Hyp	Ref	Expression
1		strcat 10664	. . . 4 |- Cat (_ ((V X. V) X. (V X. V))
2		dmss 3316	. . . 4 |- (Cat (_ ((V X. V) X. (V X. V)) -> dom Cat (_ dom ((V X. V) X. (V X. V)))
3	1, 2	ax-mp 7	. . 3 |- dom Cat (_ dom ((V X. V) X. (V X. V))
4		dmxpid 3339	. . 3 |- dom ((V X. V) X. (V X. V)) = (V X. V)
5	3, 4	sseqtr 2096	. 2 |- dom Cat (_ (V X. V)
6		df-rel 3191	. 2 |- (Rel dom Cat <-> dom Cat (_ (V X. V))
7	5, 6	mpbir 190	1 |- Rel dom Cat

Syntax hints:  Vcvv 1814   (_ wss 2050   X. cxp 3174  dom cdm 3176  Rel wrel 3181  Catccat 10656

This theorem is referenced by:  catded 10668

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-dm 3194  df-cat 10657

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