Theorem txss12 15077

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 2231	. . . 4 |- ran ( x e. B , y e. D |-> ( x X. y ) ) = ran ( x e. B , y e. D |-> ( x X. y ) )
2	1	txbasex 15068	. . 3 |- ( ( B e. V /\ D e. W ) -> ran ( x e. B , y e. D |-> ( x X. y ) ) e. _V )
3		resmpo 6129	. . . . . 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 5043	. . . . . 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 3281	. . . . 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 4968	. . . 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 14874	. . 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 599	. 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 4233	. . . . 5 |- ( ( A C_ B /\ B e. V ) -> A e. _V )
12		ssexg 4233	. . . . 5 |- ( ( C C_ D /\ D e. W ) -> C e. _V )
13		eqid 2231	. . . . . 6 |- ran ( x e. A , y e. C |-> ( x X. y ) ) = ran ( x e. A , y e. C |-> ( x X. y ) )
14	13	txval 15066	. . . . 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 592	. . 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 15066	. . 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 3273	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 1398   e. wcel 2202   _Vcvv 2803   C_ wss 3201   X. cxp 4729   ran crn 4732   |` cres 4733   ` cfv 5333  (class class class)co 6028   e. cmpo 6030   topGenctg 13417   tX ctx 15063

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-topgen 13423  df-tx 15064

This theorem is referenced by: (None)