Theorem txss12 14445

Description: Subset property of the topological product. (Contributed by Mario Carneiro, 2-Sep-2015.)

Assertion

Ref	Expression
txss12	|- ( ( ( B e. V /\ D e. W ) /\ ( A C_ B /\ C C_ D ) ) -> ( A tX C ) C_ ( B tX D ) )

Proof of Theorem txss12

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2193	. . . 4 |- ran ( x e. B , y e. D |-> ( x X. y ) ) = ran ( x e. B , y e. D |-> ( x X. y ) )
2	1	txbasex 14436	. . 3 |- ( ( B e. V /\ D e. W ) -> ran ( x e. B , y e. D |-> ( x X. y ) ) e. _V )
3		resmpo 6017	. . . . . 6 |- ( ( A C_ B /\ C C_ D ) -> ( ( x e. B , y e. D |-> ( x X. y ) ) |` ( A X. C ) ) = ( x e. A , y e. C |-> ( x X. y ) ) )
4		resss 4967	. . . . . 6 |- ( ( x e. B , y e. D |-> ( x X. y ) ) |` ( A X. C ) ) C_ ( x e. B , y e. D |-> ( x X. y ) )
5	3, 4	eqsstrrdi 3233	. . . . 5 |- ( ( A C_ B /\ C C_ D ) -> ( x e. A , y e. C |-> ( x X. y ) ) C_ ( x e. B , y e. D |-> ( x X. y ) ) )
6	5	adantl 277	. . . 4 |- ( ( ( B e. V /\ D e. W ) /\ ( A C_ B /\ C C_ D ) ) -> ( x e. A , y e. C |-> ( x X. y ) ) C_ ( x e. B , y e. D |-> ( x X. y ) ) )
7		rnss 4893	. . . 4 |- ( ( x e. A , y e. C |-> ( x X. y ) ) C_ ( x e. B , y e. D |-> ( x X. y ) ) -> ran ( x e. A , y e. C |-> ( x X. y ) ) C_ ran ( x e. B , y e. D |-> ( x X. y ) ) )
8	6, 7	syl 14	. . 3 |- ( ( ( B e. V /\ D e. W ) /\ ( A C_ B /\ C C_ D ) ) -> ran ( x e. A , y e. C |-> ( x X. y ) ) C_ ran ( x e. B , y e. D |-> ( x X. y ) ) )
9		tgss 14242	. . 3 |- ( ( ran ( x e. B , y e. D |-> ( x X. y ) ) e. _V /\ ran ( x e. A , y e. C |-> ( x X. y ) ) C_ ran ( x e. B , y e. D |-> ( x X. y ) ) ) -> ( topGen ` ran ( x e. A , y e. C |-> ( x X. y ) ) ) C_ ( topGen ` ran ( x e. B , y e. D |-> ( x X. y ) ) ) )
10	2, 8, 9	syl2an2r 595	. 2 |- ( ( ( B e. V /\ D e. W ) /\ ( A C_ B /\ C C_ D ) ) -> ( topGen ` ran ( x e. A , y e. C |-> ( x X. y ) ) ) C_ ( topGen ` ran ( x e. B , y e. D |-> ( x X. y ) ) ) )
11		ssexg 4169	. . . . 5 |- ( ( A C_ B /\ B e. V ) -> A e. _V )
12		ssexg 4169	. . . . 5 |- ( ( C C_ D /\ D e. W ) -> C e. _V )
13		eqid 2193	. . . . . 6 |- ran ( x e. A , y e. C |-> ( x X. y ) ) = ran ( x e. A , y e. C |-> ( x X. y ) )
14	13	txval 14434	. . . . 5 |- ( ( A e. _V /\ C e. _V ) -> ( A tX C ) = ( topGen ` ran ( x e. A , y e. C |-> ( x X. y ) ) ) )
15	11, 12, 14	syl2an 289	. . . 4 |- ( ( ( A C_ B /\ B e. V ) /\ ( C C_ D /\ D e. W ) ) -> ( A tX C ) = ( topGen ` ran ( x e. A , y e. C |-> ( x X. y ) ) ) )
16	15	an4s 588	. . 3 |- ( ( ( A C_ B /\ C C_ D ) /\ ( B e. V /\ D e. W ) ) -> ( A tX C ) = ( topGen ` ran ( x e. A , y e. C |-> ( x X. y ) ) ) )
17	16	ancoms 268	. 2 |- ( ( ( B e. V /\ D e. W ) /\ ( A C_ B /\ C C_ D ) ) -> ( A tX C ) = ( topGen ` ran ( x e. A , y e. C |-> ( x X. y ) ) ) )
18	1	txval 14434	. . 3 |- ( ( B e. V /\ D e. W ) -> ( B tX D ) = ( topGen ` ran ( x e. B , y e. D |-> ( x X. y ) ) ) )
19	18	adantr 276	. 2 |- ( ( ( B e. V /\ D e. W ) /\ ( A C_ B /\ C C_ D ) ) -> ( B tX D ) = ( topGen ` ran ( x e. B , y e. D |-> ( x X. y ) ) ) )
20	10, 17, 19	3sstr4d 3225	1 |- ( ( ( B e. V /\ D e. W ) /\ ( A C_ B /\ C C_ D ) ) -> ( A tX C ) C_ ( B tX D ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   _Vcvv 2760   C_ wss 3154   X. cxp 4658   ran crn 4661   |` cres 4662   ` cfv 5255  (class class class)co 5919   e. cmpo 5921   topGenctg 12868   tX ctx 14431

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-in1 615  ax-in2 616  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-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-topgen 12874  df-tx 14432

This theorem is referenced by: (None)