Theorem txss12 15290

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 2238	. . . 4 ⊢ ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦))
2	1	txbasex 15281	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) ∈ V)
3		resmpo 6176	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)))
4		resss 5082	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) ⊆ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦))
5	3, 4	eqsstrrdi 3301	. . . . 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 5007	. . . 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 15087	. . 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 603	. 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 4267	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
12		ssexg 4267	. . . . 5 ⊢ ((𝐶 ⊆ 𝐷 ∧ 𝐷 ∈ 𝑊) → 𝐶 ∈ V)
13		eqid 2238	. . . . . 6 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))
14	13	txval 15279	. . . . 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 596	. . 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 15279	. . 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 3293	1 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ⊆ wss 3220   × cxp 4767  ran crn 4770   ↾ cres 4771  ‘cfv 5372  (class class class)co 6075   ∈ cmpo 6077  topGenctg 13585   ×_t ctx 15276

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-topgen 13591  df-tx 15277

This theorem is referenced by: (None)