Theorem resixp 6763

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 4968	. . 3 |- ( F e. X_ x e. A C -> ( F |` B ) e. _V )
2	1	adantl 277	. 2 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> ( F |` B ) e. _V )
3		simpr 110	. . . . 5 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> F e. X_ x e. A C )
4		elixp2 6732	. . . . 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 122	. . . 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 1012	. . 3 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> F Fn A )
7		simpl 109	. . 3 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> B C_ A )
8		fnssres 5351	. . 3 |- ( ( F Fn A /\ B C_ A ) -> ( F |` B ) Fn B )
9	6, 7, 8	syl2anc 411	. 2 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> ( F |` B ) Fn B )
10	5	simp3d 1013	. . . 4 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> A. x e. A ( F ` x ) e. C )
11		ssralv 3234	. . . 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 5561	. . . . 5 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
14	13	eleq1d 2258	. . . 4 |- ( x e. B -> ( ( ( F |` B ) ` x ) e. C <-> ( F ` x ) e. C ) )
15	14	ralbiia 2504	. . 3 |- ( A. x e. B ( ( F |` B ) ` x ) e. C <-> A. x e. B ( F ` x ) e. C )
16	12, 15	sylibr 134	. 2 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> A. x e. B ( ( F |` B ) ` x ) e. C )
17		elixp2 6732	. 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 1183	1 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> ( F |` B ) e. X_ x e. B C )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 980   e. wcel 2160   A.wral 2468   _Vcvv 2752   C_ wss 3144   |` cres 4649   Fn wfn 5233   ` cfv 5238   X_cixp 6728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-res 4659  df-iota 5199  df-fun 5240  df-fn 5241  df-fv 5246  df-ixp 6729

This theorem is referenced by: (None)