Theorem txss12 23419

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 2724	. . . 4 ⊢ ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦))
2	1	txbasex 23380	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) ∈ V)
3		resmpo 7520	. . . . . 6 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) = (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)))
4		resss 5996	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)) ↾ (𝐴 × 𝐶)) ⊆ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦))
5	3, 4	eqsstrrdi 4029	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)))
6	5	adantl 481	. . . 4 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)) ⊆ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦)))
7		rnss 5928	. . . 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 17	. . 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 22781	. . 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 682	. 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 5313	. . . . 5 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) → 𝐴 ∈ V)
12		ssexg 5313	. . . . 5 ⊢ ((𝐶 ⊆ 𝐷 ∧ 𝐷 ∈ 𝑊) → 𝐶 ∈ V)
13		eqid 2724	. . . . . 6 ⊢ ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦)) = ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))
14	13	txval 23378	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))))
15	11, 12, 14	syl2an 595	. . . 4 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝑉) ∧ (𝐶 ⊆ 𝐷 ∧ 𝐷 ∈ 𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))))
16	15	an4s 657	. . 3 ⊢ (((𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷) ∧ (𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))))
17	16	ancoms 458	. 2 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (𝐴 ×t 𝐶) = (topGen‘ran (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ (𝑥 × 𝑦))))
18	1	txval 23378	. . 3 ⊢ ((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → (𝐵 ×t 𝐷) = (topGen‘ran (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐷 ↦ (𝑥 × 𝑦))))
19	18	adantr 480	. 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 4021	1 ⊢ (((𝐵 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) ∧ (𝐴 ⊆ 𝐵 ∧ 𝐶 ⊆ 𝐷)) → (𝐴 ×t 𝐶) ⊆ (𝐵 ×t 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  Vcvv 3466   ⊆ wss 3940   × cxp 5664  ran crn 5667   ↾ cres 5668  ‘cfv 6533  (class class class)co 7401   ∈ cmpo 7403  topGenctg 17379   ×_t ctx 23374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-topgen 17385  df-tx 23376

This theorem is referenced by: (None)