Theorem txss12 14980

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 2229	. . . 4 ⊢ ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦))
2	1	txbasex 14971	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) ∈ V)
3		resmpo 6114	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)))
4		resss 5035	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) ⊆ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦))
5	3, 4	eqsstrrdi 3278	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)))
6	5	adantl 277	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)))
7		rnss 4960	. . . 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 14777	. . 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 597	. 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 4226	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
12		ssexg 4226	. . . . 5 ⊢ ((𝐶 ⊆ 𝐷 ∧ 𝐷 ∈ 𝑊) → 𝐶 ∈ V)
13		eqid 2229	. . . . . 6 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))
14	13	txval 14969	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))))
15	11, 12, 14	syl2an 289	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐷 ∧ 𝐷 ∈ 𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))))
16	15	an4s 590	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))))
17	16	ancoms 268	. 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))))
18	1	txval 14969	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (𝐵 ×t 𝐷) = (topGen‘ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦))))
19	18	adantr 276	. 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 3270	1 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2800   ⊆ wss 3198   × cxp 4721  ran crn 4724   ↾ cres 4725  ‘cfv 5324  (class class class)co 6013   ∈ cmpo 6015  topGenctg 13327   ×_t ctx 14966

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-ov 6016  df-oprab 6017  df-mpo 6018  df-1st 6298  df-2nd 6299  df-topgen 13333  df-tx 14967

This theorem is referenced by: (None)