Theorem orderseqlem 13953

Description: Lemma for poseq 13954 and soseq 13955. The function value of a sequene is either in A or null.

Hypothesis

Ref	Expression
orderseqlem.1	|- F = {f | E.x e. On f:x-->A}

Assertion

Ref	Expression
orderseqlem	|- (G e. F -> (G` X) e. (A u. {(/)}))

Distinct variable groups:   A,f,x   f,G,x   x,X

Proof of Theorem orderseqlem

Step	Hyp	Ref	Expression
1		feq1 4551	. . . . 5 |- (f = G -> (f:x-->A <-> G:x-->A))
2	1	rexbidv 2124	. . . 4 |- (f = G -> (E.x e. On f:x-->A <-> E.x e. On G:x-->A))
3		orderseqlem.1	. . . 4 |- F = {f | E.x e. On f:x-->A}
4	2, 3	elab2g 2406	. . 3 |- (G e. F -> (G e. F <-> E.x e. On G:x-->A))
5	4	ibi 652	. 2 |- (G e. F -> E.x e. On G:x-->A)
6		ffun 4565	. . . . 5 |- (G:x-->A -> Fun G)
7		frn 4569	. . . . 5 |- (G:x-->A -> ran G C_ A)
8		unss1 2773	. . . . . . 7 |- (ran G C_ A -> (ran G u. {(/)}) C_ (A u. {(/)}))
9	8	sseld 2619	. . . . . 6 |- (ran G C_ A -> ((G` X) e. (ran G u. {(/)}) -> (G` X) e. (A u. {(/)})))
10		fvrn0 13837	. . . . . 6 |- (Fun G -> (G` X) e. (ran G u. {(/)}))
11	9, 10	syl5com 63	. . . . 5 |- (Fun G -> (ran G C_ A -> (G` X) e. (A u. {(/)})))
12	6, 7, 11	sylc 83	. . . 4 |- (G:x-->A -> (G` X) e. (A u. {(/)}))
13	12	a1i 8	. . 3 |- (x e. On -> (G:x-->A -> (G` X) e. (A u. {(/)})))
14	13	r19.23aiv 2211	. 2 |- (E.x e. On G:x-->A -> (G` X) e. (A u. {(/)}))
15	5, 14	syl 12	1 |- (G e. F -> (G` X) e. (A u. {(/)}))

Syntax hints:   -> wi 3   = wceq 1298   e. wcel 1300  {cab 1871  E.wrex 2106   u. cun 2591   C_ wss 2593  (/)c0 2875  {csn 3044  Oncon0 3657  ran crn 3987  Fun wfun 3992  -->wf 3994  ` cfv 3998

This theorem is referenced by:  poseq 13954  soseq 13955

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-13 1311  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  ax-un 3790

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-rex 2110  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-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-fv 4014

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