Theorem txuni2 12896

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 4713	. . 3 ⊢ Rel (𝑋 × 𝑌)
2		txuni2.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝑅
3	2	eleq2i 2233	. . . . . . 7 ⊢ (𝑧 ∈ 𝑋 ↔ 𝑧 ∈ ∪ 𝑅)
4		eluni2 3793	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑅 ↔ ∃𝑟 ∈ 𝑅 𝑧 ∈ 𝑟)
5	3, 4	bitri 183	. . . . . 6 ⊢ (𝑧 ∈ 𝑋 ↔ ∃𝑟 ∈ 𝑅 𝑧 ∈ 𝑟)
6		txuni2.2	. . . . . . . 8 ⊢ 𝑌 = ∪ 𝑆
7	6	eleq2i 2233	. . . . . . 7 ⊢ (𝑤 ∈ 𝑌 ↔ 𝑤 ∈ ∪ 𝑆)
8		eluni2 3793	. . . . . . 7 ⊢ (𝑤 ∈ ∪ 𝑆 ↔ ∃𝑠 ∈ 𝑆 𝑤 ∈ 𝑠)
9	7, 8	bitri 183	. . . . . 6 ⊢ (𝑤 ∈ 𝑌 ↔ ∃𝑠 ∈ 𝑆 𝑤 ∈ 𝑠)
10	5, 9	anbi12i 456	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌) ↔ (∃𝑟 ∈ 𝑅 𝑧 ∈ 𝑟 ∧ ∃𝑠 ∈ 𝑆 𝑤 ∈ 𝑠))
11		opelxp 4634	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) ↔ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌))
12		reeanv 2635	. . . . 5 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠) ↔ (∃𝑟 ∈ 𝑅 𝑧 ∈ 𝑟 ∧ ∃𝑠 ∈ 𝑆 𝑤 ∈ 𝑠))
13	10, 11, 12	3bitr4i 211	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠))
14		opelxp 4634	. . . . . 6 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠) ↔ (𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠))
15		eqid 2165	. . . . . . . . . 10 ⊢ (𝑟 × 𝑠) = (𝑟 × 𝑠)
16		xpeq1 4618	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑟 → (𝑥 × 𝑦) = (𝑟 × 𝑦))
17	16	eqeq2d 2177	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑟 → ((𝑟 × 𝑠) = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑦)))
18		xpeq2 4619	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑠 → (𝑟 × 𝑦) = (𝑟 × 𝑠))
19	18	eqeq2d 2177	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑠 → ((𝑟 × 𝑠) = (𝑟 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑠)))
20	17, 19	rspc2ev 2845	. . . . . . . . . 10 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆 ∧ (𝑟 × 𝑠) = (𝑟 × 𝑠)) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
21	15, 20	mp3an3 1316	. . . . . . . . 9 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
22		vex 2729	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
23		vex 2729	. . . . . . . . . . 11 ⊢ 𝑠 ∈ V
24	22, 23	xpex 4719	. . . . . . . . . 10 ⊢ (𝑟 × 𝑠) ∈ V
25		eqeq1 2172	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑟 × 𝑠) → (𝑧 = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑥 × 𝑦)))
26	25	2rexbidv 2491	. . . . . . . . . 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 2165	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))
29	28	rnmpo 5952	. . . . . . . . . . 11 ⊢ ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 𝑧 = (𝑥 × 𝑦)}
30	27, 29	eqtri 2186	. . . . . . . . . 10 ⊢ 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 𝑧 = (𝑥 × 𝑦)}
31	24, 26, 30	elab2 2874	. . . . . . . . 9 ⊢ ((𝑟 × 𝑠) ∈ 𝐵 ↔ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
32	21, 31	sylibr 133	. . . . . . . 8 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → (𝑟 × 𝑠) ∈ 𝐵)
33		elssuni 3817	. . . . . . . 8 ⊢ ((𝑟 × 𝑠) ∈ 𝐵 → (𝑟 × 𝑠) ⊆ ∪ 𝐵)
34	32, 33	syl 14	. . . . . . 7 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → (𝑟 × 𝑠) ⊆ ∪ 𝐵)
35	34	sseld 3141	. . . . . 6 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → (⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠) → ⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵))
36	14, 35	syl5bir 152	. . . . 5 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → ((𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠) → ⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵))
37	36	rexlimivv 2589	. . . 4 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠) → ⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵)
38	13, 37	sylbi 120	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) → ⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵)
39	1, 38	relssi 4695	. 2 ⊢ (𝑋 × 𝑌) ⊆ ∪ 𝐵
40		elssuni 3817	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑅 → 𝑥 ⊆ ∪ 𝑅)
41	40, 2	sseqtrrdi 3191	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑅 → 𝑥 ⊆ 𝑋)
42		elssuni 3817	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ⊆ ∪ 𝑆)
43	42, 6	sseqtrrdi 3191	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ⊆ 𝑌)
44		xpss12 4711	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
45	41, 43, 44	syl2an 287	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
46		vex 2729	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
47		vex 2729	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
48	46, 47	xpex 4719	. . . . . . . . 9 ⊢ (𝑥 × 𝑦) ∈ V
49	48	elpw 3565	. . . . . . . 8 ⊢ ((𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌) ↔ (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
50	45, 49	sylibr 133	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌))
51	50	rgen2 2552	. . . . . 6 ⊢ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌)
52	28	fmpo 6169	. . . . . 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 144	. . . . 5 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)):(𝑅 × 𝑆)⟶𝒫 (𝑋 × 𝑌)
54		frn 5346	. . . . 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 3174	. . 3 ⊢ 𝐵 ⊆ 𝒫 (𝑋 × 𝑌)
57		sspwuni 3950	. . 3 ⊢ (𝐵 ⊆ 𝒫 (𝑋 × 𝑌) ↔ ∪ 𝐵 ⊆ (𝑋 × 𝑌))
58	56, 57	mpbi 144	. 2 ⊢ ∪ 𝐵 ⊆ (𝑋 × 𝑌)
59	39, 58	eqssi 3158	1 ⊢ (𝑋 × 𝑌) = ∪ 𝐵

Syntax hints:   ∧ wa 103   = wceq 1343   ∈ wcel 2136  {cab 2151  ∀wral 2444  ∃wrex 2445   ⊆ wss 3116  𝒫 cpw 3559  ⟨cop 3579  ∪ cuni 3789   × cxp 4602  ran crn 4605  ⟶wf 5184   ∈ cmpo 5844

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-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-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  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-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  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-fv 5196  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109

This theorem is referenced by:  txbasex  12897  txtopon  12902