Theorem resixp 7407

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 5256	. . 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 7376	. . . . 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 5631	. . 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 3523	. . . 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 5812	. . . . 5 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
14	13	eleq1d 2523	. . . 4 |- ( x e. B -> ( ( ( F |` B ) ` x ) e. C <-> ( F ` x ) e. C ) )
15	14	ralbiia 2837	. . 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 7376	. 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 1758   A.wral 2798   _Vcvv 3076   C_ wss 3435   |` cres 4949   Fn wfn 5520   ` cfv 5525   X_cixp 7372

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-br 4400  df-opab 4458  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-res 4959  df-iota 5488  df-fun 5527  df-fn 5528  df-fv 5533  df-ixp 7373

This theorem is referenced by:  resixpfo  7410  ixpfi2  7719  ptrescn  19343  ptuncnv  19511  ptcmplem2  19756