Theorem relrcat 10667

Description: The range of a category is a relation.

Assertion

Ref	Expression
relrcat	|- Rel ran Cat

Proof of Theorem relrcat

Step	Hyp	Ref	Expression
1		strcat 10664	. . . 4 |- Cat (_ ((V X. V) X. (V X. V))
2		rnss 3348	. . . 4 |- (Cat (_ ((V X. V) X. (V X. V)) -> ran Cat (_ ran ((V X. V) X. (V X. V)))
3	1, 2	ax-mp 7	. . 3 |- ran Cat (_ ran ((V X. V) X. (V X. V))
4		vne0 2290	. . . . . 6 |- V =/= (/)
5	4, 4	pm3.2i 285	. . . . 5 |- (V =/= (/) /\ V =/= (/))
6		xpnz 3472	. . . . 5 |- ((V =/= (/) /\ V =/= (/)) <-> (V X. V) =/= (/))
7	5, 6	mpbi 189	. . . 4 |- (V X. V) =/= (/)
8		rnxp 3478	. . . 4 |- ((V X. V) =/= (/) -> ran ((V X. V) X. (V X. V)) = (V X. V))
9	7, 8	ax-mp 7	. . 3 |- ran ((V X. V) X. (V X. V)) = (V X. V)
10	3, 9	sseqtr 2096	. 2 |- ran Cat (_ (V X. V)
11		df-rel 3191	. 2 |- (Rel ran Cat <-> ran Cat (_ (V X. V))
12	10, 11	mpbir 190	1 |- Rel ran Cat

Syntax hints:   /\ wa 223   = wceq 958   =/= wne 1588  Vcvv 1814   (_ wss 2050  (/)c0 2283   X. cxp 3174  ran crn 3177  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-cnv 3192  df-dm 3194  df-rn 3195  df-cat 10657

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