Theorem relrded 10675

Description: The range of a deductive system is a relation.

Assertion

Ref	Expression
relrded	|- Rel ran Ded

Proof of Theorem relrded

Step	Hyp	Ref	Expression
1		strded 10672	. . . 4 |- Ded (_ ((V X. V) X. (V X. V))
2		rnss 3342	. . . 4 |- (Ded (_ ((V X. V) X. (V X. V)) -> ran Ded (_ ran ((V X. V) X. (V X. V)))
3	1, 2	ax-mp 7	. . 3 |- ran Ded (_ ran ((V X. V) X. (V X. V))
4		0ex 2711	. . . . . . 7 |- (/) e. V
5		ne0i 2286	. . . . . . 7 |- ((/) e. V -> V =/= (/))
6	4, 5	ax-mp 7	. . . . . 6 |- V =/= (/)
7	6, 6	pm3.2i 285	. . . . 5 |- (V =/= (/) /\ V =/= (/))
8		xpnz 3466	. . . . 5 |- ((V =/= (/) /\ V =/= (/)) <-> (V X. V) =/= (/))
9	7, 8	mpbi 189	. . . 4 |- (V X. V) =/= (/)
10		rnxp 3472	. . . 4 |- ((V X. V) =/= (/) -> ran ((V X. V) X. (V X. V)) = (V X. V))
11	9, 10	ax-mp 7	. . 3 |- ran ((V X. V) X. (V X. V)) = (V X. V)
12	3, 11	sseqtr 2093	. 2 |- ran Ded (_ (V X. V)
13		df-rel 3185	. 2 |- (Rel ran Ded <-> ran Ded (_ (V X. V))
14	12, 13	mpbir 190	1 |- Rel ran Ded

Syntax hints:   /\ wa 223   = wceq 956   e. wcel 958   =/= wne 1585  Vcvv 1811   (_ wss 2047  (/)c0 2280   X. cxp 3168  ran crn 3171  Rel wrel 3175  Dedcded 10667

This theorem is referenced by:  dedalg 10676

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-br 2620  df-opab 2667  df-xp 3184  df-rel 3185  df-cnv 3186  df-dm 3188  df-rn 3189  df-ded 10668

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