Theorem resixp 7290

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 5144	. . 3 |- ( F e. X_ x e. A C -> ( F |` B ) e. _V )
2	1	adantl 466	. 2 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> ( F |` B ) e. _V )
3		simpr 461	. . . . 5 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> F e. X_ x e. A C )
4		elixp2 7259	. . . . 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 196	. . . 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 1001	. . 3 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> F Fn A )
7		simpl 457	. . 3 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> B C_ A )
8		fnssres 5519	. . 3 |- ( ( F Fn A /\ B C_ A ) -> ( F |` B ) Fn B )
9	6, 7, 8	syl2anc 661	. 2 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> ( F |` B ) Fn B )
10	5	simp3d 1002	. . . 4 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> A. x e. A ( F ` x ) e. C )
11		ssralv 3411	. . . 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 60	. . 3 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> A. x e. B ( F ` x ) e. C )
13		fvres 5699	. . . . 5 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
14	13	eleq1d 2504	. . . 4 |- ( x e. B -> ( ( ( F |` B ) ` x ) e. C <-> ( F ` x ) e. C ) )
15	14	ralbiia 2742	. . 3 |- ( A. x e. B ( ( F |` B ) ` x ) e. C <-> A. x e. B ( F ` x ) e. C )
16	12, 15	sylibr 212	. 2 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> A. x e. B ( ( F |` B ) ` x ) e. C )
17		elixp2 7259	. 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 1172	1 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> ( F |` B ) e. X_ x e. B C )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   e. wcel 1756   A.wral 2710   _Vcvv 2967   C_ wss 3323   |` cres 4837   Fn wfn 5408   ` cfv 5413   X_cixp 7255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-res 4847  df-iota 5376  df-fun 5415  df-fn 5416  df-fv 5421  df-ixp 7256

This theorem is referenced by:  resixpfo  7293  ixpfi2  7601  ptrescn  19187  ptuncnv  19355  ptcmplem2  19600