Theorem orderseqlem 31749

Description: Lemma for poseq 31750 and soseq 31751. The function value of a sequene is either in A or null. (Contributed by Scott Fenton, 8-Jun-2011.)

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

Allowed substitution hints:   F( x, f)   X( f)

Proof of Theorem orderseqlem

Step	Hyp	Ref	Expression
1		feq1 6026	. . . . 5 |- ( f = G -> ( f : x --> A <-> G : x --> A ) )
2	1	rexbidv 3052	. . . 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 3353	. . 3 |- ( G e. F -> ( G e. F <-> E. x e. On G : x --> A ) )
5	4	ibi 256	. 2 |- ( G e. F -> E. x e. On G : x --> A )
6		frn 6053	. . . . 5 |- ( G : x --> A -> ran G C_ A )
7		unss1 3782	. . . . 5 |- ( ran G C_ A -> ( ran G u. { (/) } ) C_ ( A u. { (/) } ) )
8	6, 7	syl 17	. . . 4 |- ( G : x --> A -> ( ran G u. { (/) } ) C_ ( A u. { (/) } ) )
9		fvrn0 6216	. . . 4 |- ( G ` X ) e. ( ran G u. { (/) } )
10		ssel 3597	. . . 4 |- ( ( ran G u. { (/) } ) C_ ( A u. { (/) } ) -> ( ( G ` X ) e. ( ran G u. { (/) } ) -> ( G ` X ) e. ( A u. { (/) } ) ) )
11	8, 9, 10	mpisyl 21	. . 3 |- ( G : x --> A -> ( G ` X ) e. ( A u. { (/) } ) )
12	11	rexlimivw 3029	. 2 |- ( E. x e. On G : x --> A -> ( G ` X ) e. ( A u. { (/) } ) )
13	5, 12	syl 17	1 |- ( G e. F -> ( G ` X ) e. ( A u. { (/) } ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {cab 2608   E.wrex 2913   u. cun 3572   C_ wss 3574   (/)c0 3915   {csn 4177   ran crn 5115   Oncon0 5723   -->wf 5884   ` cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  poseq  31750  soseq  31751