Theorem resixp 6901

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 5053	. . 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 6870	. . . . 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 1036	. . 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 5445	. . 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 1037	. . . 4 |- ( ( B C_ A /\ F e. X_ x e. A C ) -> A. x e. A ( F ` x ) e. C )
11		ssralv 3291	. . . 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 5663	. . . . 5 |- ( x e. B -> ( ( F |` B ) ` x ) = ( F ` x ) )
14	13	eleq1d 2300	. . . 4 |- ( x e. B -> ( ( ( F |` B ) ` x ) e. C <-> ( F ` x ) e. C ) )
15	14	ralbiia 2546	. . 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 6870	. 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 1207	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 1004   e. wcel 2202   A.wral 2510   _Vcvv 2802   C_ wss 3200   |` cres 4727   Fn wfn 5321   ` cfv 5326   X_cixp 6866

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-fv 5334  df-ixp 6867

This theorem is referenced by: (None)