Theorem resixp 7565

Description: Restriction of an element of an infinite Cartesian product. (Contributed by FL, 7-Nov-2011.) (Proof shortened by Mario Carneiro, 31-May-2014.)

Assertion

Ref	Expression
resixp	|- ( ( B C_ A /\ F e. X_ x e. A C ) -> ( F |` B ) e. X_ x e. B C )

Distinct variable groups:   x, A   x, B   x, F

Allowed substitution hint:   C( x)

Proof of Theorem resixp

Step	Hyp	Ref	Expression
1		resexg 5167	. . 3 |- ( F e. X_ x e. A C -> ( F |` B ) e. _V )
2	1	adantl 467	. 2 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> ( F |` B ) e. _V )
3		simpr 462	. . . . 5 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> F e. X_ x e. A C )
4		elixp2 7534	. . . . 5 |- ( F e. X_ x e. A C <-> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. C ) )
5	3, 4	sylib 199	. . . 4 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> ( F e. _V /\ F Fn A /\ A. x e. A ( F ` x ) e. C ) )
6	5	simp2d 1018	. . 3 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> F Fn A )
7		simpl 458	. . 3 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> B C_ A )
8		fnssres 5707	. . 3 |- ( ( F Fn A /\ B C_ A ) -> ( F |` B ) Fn B )
9	6, 7, 8	syl2anc 665	. 2 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> ( F |` B ) Fn B )
10	5	simp3d 1019	. . . 4 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> A. x e. A ( F ` x ) e. C )
11		ssralv 3531	. . . 4 |- ( B C_ A -> ( A. x e. A ( F ` x ) e. C -> A. x e. B ( F ` x ) e. C ) )
12	7, 10, 11	sylc 62	. . 3 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> A. x e. B ( F ` x ) e. C )
13		fvres 5895	. . . . 5 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
14	13	eleq1d 2498	. . . 4 |- ( x e. B -> ( ( ( F |` B ) ` x ) e. C <-> ( F ` x ) e. C ) )
15	14	ralbiia 2862	. . 3 |- ( A. x e. B ( ( F |` B ) ` x ) e. C <-> A. x e. B ( F ` x ) e. C )
16	12, 15	sylibr 215	. 2 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> A. x e. B ( ( F |` B ) ` x ) e. C )
17		elixp2 7534	. 2 |- ( ( F |` B ) e. X_ x e. B C <-> ( ( F |` B ) e. _V /\ ( F |` B ) Fn B /\ A. x e. B ( ( F |` B ) ` x ) e. C ) )
18	2, 9, 16, 17	syl3anbrc 1189	1 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> ( F |` B ) e. X_ x e. B C )

Syntax hints:   -> wi 4   /\ wa 370   /\ w3a 982   e. wcel 1870   A.wral 2782   _Vcvv 3087   C_ wss 3442   |` cres 4856   Fn wfn 5596   ` cfv 5601   X_cixp 7530

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-res 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-fv 5609  df-ixp 7531

This theorem is referenced by:  resixpfo  7568  ixpfi2  7878  ptrescn  20585  ptuncnv  20753  ptcmplem2  20999