Theorem txuni2 23480

Description: The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 31-Aug-2015.)

Hypotheses

Ref	Expression
txval.1	⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))
txuni2.1	⊢ 𝑋 = ∪ 𝑅
txuni2.2	⊢ 𝑌 = ∪ 𝑆

Assertion

Ref	Expression
txuni2	⊢ (𝑋 × 𝑌) = ∪ 𝐵

Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝑆,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦)

Proof of Theorem txuni2

Dummy variables 𝑟 𝑠 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relxp 5632	. . 3 ⊢ Rel (𝑋 × 𝑌)
2		txuni2.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝑅
3	2	eleq2i 2823	. . . . . . 7 ⊢ (𝑧 ∈ 𝑋 ↔ 𝑧 ∈ ∪ 𝑅)
4		eluni2 4860	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑅 ↔ ∃𝑟 ∈ 𝑅 𝑧 ∈ 𝑟)
5	3, 4	bitri 275	. . . . . 6 ⊢ (𝑧 ∈ 𝑋 ↔ ∃𝑟 ∈ 𝑅 𝑧 ∈ 𝑟)
6		txuni2.2	. . . . . . . 8 ⊢ 𝑌 = ∪ 𝑆
7	6	eleq2i 2823	. . . . . . 7 ⊢ (𝑤 ∈ 𝑌 ↔ 𝑤 ∈ ∪ 𝑆)
8		eluni2 4860	. . . . . . 7 ⊢ (𝑤 ∈ ∪ 𝑆 ↔ ∃𝑠 ∈ 𝑆 𝑤 ∈ 𝑠)
9	7, 8	bitri 275	. . . . . 6 ⊢ (𝑤 ∈ 𝑌 ↔ ∃𝑠 ∈ 𝑆 𝑤 ∈ 𝑠)
10	5, 9	anbi12i 628	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌) ↔ (∃𝑟 ∈ 𝑅 𝑧 ∈ 𝑟 ∧ ∃𝑠 ∈ 𝑆 𝑤 ∈ 𝑠))
11		opelxp 5650	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) ↔ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌))
12		reeanv 3204	. . . . 5 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠) ↔ (∃𝑟 ∈ 𝑅 𝑧 ∈ 𝑟 ∧ ∃𝑠 ∈ 𝑆 𝑤 ∈ 𝑠))
13	10, 11, 12	3bitr4i 303	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠))
14		opelxp 5650	. . . . . 6 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠) ↔ (𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠))
15		eqid 2731	. . . . . . . . . 10 ⊢ (𝑟 × 𝑠) = (𝑟 × 𝑠)
16		xpeq1 5628	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑟 → (𝑥 × 𝑦) = (𝑟 × 𝑦))
17	16	eqeq2d 2742	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑟 → ((𝑟 × 𝑠) = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑦)))
18		xpeq2 5635	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑠 → (𝑟 × 𝑦) = (𝑟 × 𝑠))
19	18	eqeq2d 2742	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑠 → ((𝑟 × 𝑠) = (𝑟 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑠)))
20	17, 19	rspc2ev 3585	. . . . . . . . . 10 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆 ∧ (𝑟 × 𝑠) = (𝑟 × 𝑠)) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
21	15, 20	mp3an3 1452	. . . . . . . . 9 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
22		vex 3440	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
23		vex 3440	. . . . . . . . . . 11 ⊢ 𝑠 ∈ V
24	22, 23	xpex 7686	. . . . . . . . . 10 ⊢ (𝑟 × 𝑠) ∈ V
25		eqeq1 2735	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑟 × 𝑠) → (𝑧 = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑥 × 𝑦)))
26	25	2rexbidv 3197	. . . . . . . . . 10 ⊢ (𝑧 = (𝑟 × 𝑠) → (∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 𝑧 = (𝑥 × 𝑦) ↔ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦)))
27		txval.1	. . . . . . . . . . 11 ⊢ 𝐵 = ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))
28		eqid 2731	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))
29	28	rnmpo 7479	. . . . . . . . . . 11 ⊢ ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 𝑧 = (𝑥 × 𝑦)}
30	27, 29	eqtri 2754	. . . . . . . . . 10 ⊢ 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 𝑧 = (𝑥 × 𝑦)}
31	24, 26, 30	elab2 3633	. . . . . . . . 9 ⊢ ((𝑟 × 𝑠) ∈ 𝐵 ↔ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
32	21, 31	sylibr 234	. . . . . . . 8 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → (𝑟 × 𝑠) ∈ 𝐵)
33		elssuni 4887	. . . . . . . 8 ⊢ ((𝑟 × 𝑠) ∈ 𝐵 → (𝑟 × 𝑠) ⊆ ∪ 𝐵)
34	32, 33	syl 17	. . . . . . 7 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → (𝑟 × 𝑠) ⊆ ∪ 𝐵)
35	34	sseld 3928	. . . . . 6 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → (⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠) → ⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵))
36	14, 35	biimtrrid 243	. . . . 5 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → ((𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠) → ⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵))
37	36	rexlimivv 3174	. . . 4 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠) → ⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵)
38	13, 37	sylbi 217	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) → ⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵)
39	1, 38	relssi 5726	. 2 ⊢ (𝑋 × 𝑌) ⊆ ∪ 𝐵
40		elssuni 4887	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑅 → 𝑥 ⊆ ∪ 𝑅)
41	40, 2	sseqtrrdi 3971	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑅 → 𝑥 ⊆ 𝑋)
42		elssuni 4887	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ⊆ ∪ 𝑆)
43	42, 6	sseqtrrdi 3971	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ⊆ 𝑌)
44		xpss12 5629	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
45	41, 43, 44	syl2an 596	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
46		vex 3440	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
47		vex 3440	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
48	46, 47	xpex 7686	. . . . . . . . 9 ⊢ (𝑥 × 𝑦) ∈ V
49	48	elpw 4551	. . . . . . . 8 ⊢ ((𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌) ↔ (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
50	45, 49	sylibr 234	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌))
51	50	rgen2 3172	. . . . . 6 ⊢ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌)
52	28	fmpo 8000	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌) ↔ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)):(𝑅 × 𝑆)⟶𝒫 (𝑋 × 𝑌))
53	51, 52	mpbi 230	. . . . 5 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)):(𝑅 × 𝑆)⟶𝒫 (𝑋 × 𝑌)
54		frn 6658	. . . . 5 ⊢ ((𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)):(𝑅 × 𝑆)⟶𝒫 (𝑋 × 𝑌) → ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) ⊆ 𝒫 (𝑋 × 𝑌))
55	53, 54	ax-mp 5	. . . 4 ⊢ ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) ⊆ 𝒫 (𝑋 × 𝑌)
56	27, 55	eqsstri 3976	. . 3 ⊢ 𝐵 ⊆ 𝒫 (𝑋 × 𝑌)
57		sspwuni 5046	. . 3 ⊢ (𝐵 ⊆ 𝒫 (𝑋 × 𝑌) ↔ ∪ 𝐵 ⊆ (𝑋 × 𝑌))
58	56, 57	mpbi 230	. 2 ⊢ ∪ 𝐵 ⊆ (𝑋 × 𝑌)
59	39, 58	eqssi 3946	1 ⊢ (𝑋 × 𝑌) = ∪ 𝐵

Syntax hints:   ∧ wa 395   = wceq 1541   ∈ wcel 2111  {cab 2709  ∀wral 3047  ∃wrex 3056   ⊆ wss 3897  𝒫 cpw 4547  ⟨cop 4579  ∪ cuni 4856   × cxp 5612  ran crn 5615  ⟶wf 6477   ∈ cmpo 7348

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922

This theorem is referenced by:  txbasex  23481  txtopon  23506  sxsigon  34205