Theorem relrded 10626

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 10623	. . . 4 |- Ded (_ ((V X. V) X. (V X. V))
2		rnss 3339	. . . 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 2708	. . . . . . 7 |- (/) e. V
5		ne0i 2284	. . . . . . 7 |- ((/) e. V -> V =/= (/))
6	4, 5	ax-mp 7	. . . . . 6 |- V =/= (/)
7	6, 6	pm3.2i 285	. . . . 5 |- (V =/= (/) /\ V =/= (/))
8		xpnz 3463	. . . . 5 |- ((V =/= (/) /\ V =/= (/)) <-> (V X. V) =/= (/))
9	7, 8	mpbi 189	. . . 4 |- (V X. V) =/= (/)
10		rnxp 3469	. . . 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 2091	. 2 |- ran Ded (_ (V X. V)
13		df-rel 3182	. 2 |- (Rel ran Ded <-> ran Ded (_ (V X. V))
14	12, 13	mpbir 190	1 |- Rel ran Ded

Syntax hints:   /\ wa 223   = wceq 955   e. wcel 957   =/= wne 1584  Vcvv 1809   (_ wss 2045  (/)c0 2278   X. cxp 3165  ran crn 3168  Rel wrel 3172  Dedcded 10618

This theorem is referenced by:  dedalg 10627

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2700  ax-nul 2707  ax-pow 2739  ax-pr 2776

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-ral 1648  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-br 2617  df-opab 2664  df-xp 3181  df-rel 3182  df-cnv 3183  df-dm 3185  df-rn 3186  df-ded 10619

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