Theorem txss12 14853

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 2207	. . . 4 |- ran ( x e. B , y e. D |-> ( x X. y ) ) = ran ( x e. B , y e. D |-> ( x X. y ) )
2	1	txbasex 14844	. . 3 |- ( ( B e. V /\ D e. W ) -> ran ( x e. B , y e. D |-> ( x X. y ) ) e. _V )
3		resmpo 6066	. . . . . 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 5002	. . . . . 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 3254	. . . . 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 4927	. . . 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 14650	. . 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 4199	. . . . 5 |- ( ( A C_ B /\ B e. V ) -> A e. _V )
12		ssexg 4199	. . . . 5 |- ( ( C C_ D /\ D e. W ) -> C e. _V )
13		eqid 2207	. . . . . 6 |- ran ( x e. A , y e. C |-> ( x X. y ) ) = ran ( x e. A , y e. C |-> ( x X. y ) )
14	13	txval 14842	. . . . 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 14842	. . 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 3246	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 1373   e. wcel 2178   _Vcvv 2776   C_ wss 3174   X. cxp 4691   ran crn 4694   |` cres 4695   ` cfv 5290  (class class class)co 5967   e. cmpo 5969   topGenctg 13201   tX ctx 14839

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-topgen 13207  df-tx 14840

This theorem is referenced by: (None)