Theorem txss12 12906

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

Assertion

Ref	Expression
txss12	⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))

Proof of Theorem txss12

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

Step	Hyp	Ref	Expression
1		eqid 2165	. . . 4 ⊢ ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦))
2	1	txbasex 12897	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) ∈ V)
3		resmpo 5940	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)))
4		resss 4908	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) ⊆ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦))
5	3, 4	eqsstrrdi 3195	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)))
6	5	adantl 275	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)))
7		rnss 4834	. . . 4 ⊢ ((𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) → ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)) ⊆ ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)))
8	6, 7	syl 14	. . 3 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)) ⊆ ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)))
9		tgss 12703	. . 3 ⊢ ((ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) ∈ V ∧ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)) ⊆ ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦))) → (topGen‘ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))) ⊆ (topGen‘ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦))))
10	2, 8, 9	syl2an2r 585	. 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (topGen‘ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))) ⊆ (topGen‘ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦))))
11		ssexg 4121	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
12		ssexg 4121	. . . . 5 ⊢ ((𝐶 ⊆ 𝐷 ∧ 𝐷 ∈ 𝑊) → 𝐶 ∈ V)
13		eqid 2165	. . . . . 6 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))
14	13	txval 12895	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))))
15	11, 12, 14	syl2an 287	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐷 ∧ 𝐷 ∈ 𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))))
16	15	an4s 578	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))))
17	16	ancoms 266	. 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))))
18	1	txval 12895	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (𝐵 ×t 𝐷) = (topGen‘ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦))))
19	18	adantr 274	. 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (𝐵 ×t 𝐷) = (topGen‘ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦))))
20	10, 17, 19	3sstr4d 3187	1 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116   × cxp 4602  ran crn 4605   ↾ cres 4606  ‘cfv 5188  (class class class)co 5842   ∈ cmpo 5844  topGenctg 12571   ×_t ctx 12892

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-topgen 12577  df-tx 12893

This theorem is referenced by: (None)