Theorem bnd2lem 33590

Description: Lemma for equivbnd2 33591 and similar theorems. (Contributed by Jeff Madsen, 16-Sep-2015.)

Hypothesis

Ref	Expression
bnd2lem.1	|- D = ( M |` ( Y X. Y ) )

Assertion

Ref	Expression
bnd2lem	|- ( ( M e. ( Met ` X ) /\ D e. ( Bnd ` Y ) ) -> Y C_ X )

Proof of Theorem bnd2lem

Step	Hyp	Ref	Expression
1		bnd2lem.1	. . . . . 6 |- D = ( M |` ( Y X. Y ) )
2		resss 5422	. . . . . 6 |- ( M |` ( Y X. Y ) ) C_ M
3	1, 2	eqsstri 3635	. . . . 5 |- D C_ M
4		dmss 5323	. . . . 5 |- ( D C_ M -> dom D C_ dom M )
5	3, 4	mp1i 13	. . . 4 |- ( ( M e. ( Met ` X ) /\ D e. ( Bnd ` Y ) ) -> dom D C_ dom M )
6		bndmet 33580	. . . . . 6 |- ( D e. ( Bnd ` Y ) -> D e. ( Met ` Y ) )
7		metf 22135	. . . . . 6 |- ( D e. ( Met ` Y ) -> D : ( Y X. Y ) --> RR )
8		fdm 6051	. . . . . 6 |- ( D : ( Y X. Y ) --> RR -> dom D = ( Y X. Y ) )
9	6, 7, 8	3syl 18	. . . . 5 |- ( D e. ( Bnd ` Y ) -> dom D = ( Y X. Y ) )
10	9	adantl 482	. . . 4 |- ( ( M e. ( Met ` X ) /\ D e. ( Bnd ` Y ) ) -> dom D = ( Y X. Y ) )
11		metf 22135	. . . . . 6 |- ( M e. ( Met ` X ) -> M : ( X X. X ) --> RR )
12		fdm 6051	. . . . . 6 |- ( M : ( X X. X ) --> RR -> dom M = ( X X. X ) )
13	11, 12	syl 17	. . . . 5 |- ( M e. ( Met ` X ) -> dom M = ( X X. X ) )
14	13	adantr 481	. . . 4 |- ( ( M e. ( Met ` X ) /\ D e. ( Bnd ` Y ) ) -> dom M = ( X X. X ) )
15	5, 10, 14	3sstr3d 3647	. . 3 |- ( ( M e. ( Met ` X ) /\ D e. ( Bnd ` Y ) ) -> ( Y X. Y ) C_ ( X X. X ) )
16		dmss 5323	. . 3 |- ( ( Y X. Y ) C_ ( X X. X ) -> dom ( Y X. Y ) C_ dom ( X X. X ) )
17	15, 16	syl 17	. 2 |- ( ( M e. ( Met ` X ) /\ D e. ( Bnd ` Y ) ) -> dom ( Y X. Y ) C_ dom ( X X. X ) )
18		dmxpid 5345	. 2 |- dom ( Y X. Y ) = Y
19		dmxpid 5345	. 2 |- dom ( X X. X ) = X
20	17, 18, 19	3sstr3g 3645	1 |- ( ( M e. ( Met ` X ) /\ D e. ( Bnd ` Y ) ) -> Y C_ X )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   C_ wss 3574   X. cxp 5112   dom cdm 5114   |` cres 5116   -->wf 5884   ` cfv 5888   RRcr 9935   Metcme 19732   Bndcbnd 33566

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-met 19740  df-bnd 33578

This theorem is referenced by:  equivbnd2  33591  prdsbnd2  33594  cntotbnd  33595  cnpwstotbnd  33596