Theorem txss12 13060

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 2170	. . . 4 |- ran ( x e. B , y e. D |-> ( x X. y ) ) = ran ( x e. B , y e. D |-> ( x X. y ) )
2	1	txbasex 13051	. . 3 |- ( ( B e. V /\ D e. W ) -> ran ( x e. B , y e. D |-> ( x X. y ) ) e. _V )
3		resmpo 5951	. . . . . 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 4915	. . . . . 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 3200	. . . . 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 275	. . . 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 4841	. . . 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 12857	. . 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 590	. 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 4128	. . . . 5 |- ( ( A C_ B /\ B e. V ) -> A e. _V )
12		ssexg 4128	. . . . 5 |- ( ( C C_ D /\ D e. W ) -> C e. _V )
13		eqid 2170	. . . . . 6 |- ran ( x e. A , y e. C |-> ( x X. y ) ) = ran ( x e. A , y e. C |-> ( x X. y ) )
14	13	txval 13049	. . . . 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 287	. . . 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 583	. . 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 266	. 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 13049	. . 3 |- ( ( B e. V /\ D e. W ) -> ( B tX D ) = ( topGen ` ran ( x e. B , y e. D |-> ( x X. y ) ) ) )
19	18	adantr 274	. 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 3192	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 103   = wceq 1348   e. wcel 2141   _Vcvv 2730   C_ wss 3121   X. cxp 4609   ran crn 4612   |` cres 4613   ` cfv 5198  (class class class)co 5853   e. cmpo 5855   topGenctg 12594   tX ctx 13046

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-topgen 12600  df-tx 13047

This theorem is referenced by: (None)