Theorem relrded 15089

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 15086	. . . 4 |- Ded C_ ((_V X. _V) X. (_V X. _V))
2		rnss 4189	. . . 4 |- ( Ded C_ ((_V X. _V) X. (_V X. _V)) -> ran Ded C_ ran ((_V X. _V) X. (_V X. _V)))
3	1, 2	ax-mp 7	. . 3 |- ran Ded C_ ran ((_V X. _V) X. (_V X. _V))
4		vn0 2882	. . . . . 6 |- _V =/= (/)
5	4, 4	pm3.2i 307	. . . . 5 |- (_V =/= (/) /\ _V =/= (/))
6		xpnz 4335	. . . . 5 |- ((_V =/= (/) /\ _V =/= (/)) <-> (_V X. _V) =/= (/))
7	5, 6	mpbi 206	. . . 4 |- (_V X. _V) =/= (/)
8		rnxp 4342	. . . 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	sseqtri 2649	. 2 |- ran Ded C_ (_V X. _V)
11		df-rel 4001	. 2 |- (Rel ran Ded <-> ran Ded C_ (_V X. _V))
12	10, 11	mpbir 207	1 |- Rel ran Ded

Syntax hints:   /\ wa 240   = wceq 1298   =/= wne 2017  _Vcvv 2292   C_ wss 2593  (/)c0 2875   X. cxp 3984  ran crn 3987  Rel wrel 3991   Ded cded 15081

This theorem is referenced by:  dedalg 15090

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-cnv 4002  df-dm 4004  df-rn 4005  df-ded 15082

Copyright terms: Public domain