Theorem txuni2 15247

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 4864	. . 3 ⊢ Rel (𝑋 × 𝑌)
2		txuni2.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝑅
3	2	eleq2i 2301	. . . . . . 7 ⊢ (𝑧 ∈ 𝑋 ↔ 𝑧 ∈ ∪ 𝑅)
4		eluni2 3923	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑅 ↔ ∃𝑟 ∈ 𝑅 𝑧 ∈ 𝑟)
5	3, 4	bitri 184	. . . . . 6 ⊢ (𝑧 ∈ 𝑋 ↔ ∃𝑟 ∈ 𝑅 𝑧 ∈ 𝑟)
6		txuni2.2	. . . . . . . 8 ⊢ 𝑌 = ∪ 𝑆
7	6	eleq2i 2301	. . . . . . 7 ⊢ (𝑤 ∈ 𝑌 ↔ 𝑤 ∈ ∪ 𝑆)
8		eluni2 3923	. . . . . . 7 ⊢ (𝑤 ∈ ∪ 𝑆 ↔ ∃𝑠 ∈ 𝑆 𝑤 ∈ 𝑠)
9	7, 8	bitri 184	. . . . . 6 ⊢ (𝑤 ∈ 𝑌 ↔ ∃𝑠 ∈ 𝑆 𝑤 ∈ 𝑠)
10	5, 9	anbi12i 460	. . . . 5 ⊢ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌) ↔ (∃𝑟 ∈ 𝑅 𝑧 ∈ 𝑟 ∧ ∃𝑠 ∈ 𝑆 𝑤 ∈ 𝑠))
11		opelxp 4784	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) ↔ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌))
12		reeanv 2715	. . . . 5 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠) ↔ (∃𝑟 ∈ 𝑅 𝑧 ∈ 𝑟 ∧ ∃𝑠 ∈ 𝑆 𝑤 ∈ 𝑠))
13	10, 11, 12	3bitr4i 212	. . . 4 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) ↔ ∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠))
14		opelxp 4784	. . . . . 6 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠) ↔ (𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠))
15		eqid 2234	. . . . . . . . . 10 ⊢ (𝑟 × 𝑠) = (𝑟 × 𝑠)
16		xpeq1 4768	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑟 → (𝑥 × 𝑦) = (𝑟 × 𝑦))
17	16	eqeq2d 2246	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑟 → ((𝑟 × 𝑠) = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑦)))
18		xpeq2 4769	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑠 → (𝑟 × 𝑦) = (𝑟 × 𝑠))
19	18	eqeq2d 2246	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑠 → ((𝑟 × 𝑠) = (𝑟 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑠)))
20	17, 19	rspc2ev 2939	. . . . . . . . . 10 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆 ∧ (𝑟 × 𝑠) = (𝑟 × 𝑠)) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
21	15, 20	mp3an3 1363	. . . . . . . . 9 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
22		vex 2818	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
23		vex 2818	. . . . . . . . . . 11 ⊢ 𝑠 ∈ V
24	22, 23	xpex 4871	. . . . . . . . . 10 ⊢ (𝑟 × 𝑠) ∈ V
25		eqeq1 2241	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑟 × 𝑠) → (𝑧 = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑥 × 𝑦)))
26	25	2rexbidv 2569	. . . . . . . . . 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 2234	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))
29	28	rnmpo 6172	. . . . . . . . . . 11 ⊢ ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 𝑧 = (𝑥 × 𝑦)}
30	27, 29	eqtri 2255	. . . . . . . . . 10 ⊢ 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 𝑧 = (𝑥 × 𝑦)}
31	24, 26, 30	elab2 2968	. . . . . . . . 9 ⊢ ((𝑟 × 𝑠) ∈ 𝐵 ↔ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
32	21, 31	sylibr 134	. . . . . . . 8 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → (𝑟 × 𝑠) ∈ 𝐵)
33		elssuni 3947	. . . . . . . 8 ⊢ ((𝑟 × 𝑠) ∈ 𝐵 → (𝑟 × 𝑠) ⊆ ∪ 𝐵)
34	32, 33	syl 14	. . . . . . 7 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → (𝑟 × 𝑠) ⊆ ∪ 𝐵)
35	34	sseld 3241	. . . . . 6 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → (⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠) → ⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵))
36	14, 35	biimtrrid 153	. . . . 5 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → ((𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠) → ⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵))
37	36	rexlimivv 2668	. . . 4 ⊢ (∃𝑟 ∈ 𝑅 ∃𝑠 ∈ 𝑆 (𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠) → ⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵)
38	13, 37	sylbi 121	. . 3 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) → ⟨𝑧, 𝑤⟩ ∈ ∪ 𝐵)
39	1, 38	relssi 4846	. 2 ⊢ (𝑋 × 𝑌) ⊆ ∪ 𝐵
40		elssuni 3947	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑅 → 𝑥 ⊆ ∪ 𝑅)
41	40, 2	sseqtrrdi 3291	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑅 → 𝑥 ⊆ 𝑋)
42		elssuni 3947	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ⊆ ∪ 𝑆)
43	42, 6	sseqtrrdi 3291	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ⊆ 𝑌)
44		xpss12 4862	. . . . . . . . 9 ⊢ ((𝑥 ⊆ 𝑋 ∧ 𝑦 ⊆ 𝑌) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
45	41, 43, 44	syl2an 289	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
46		vex 2818	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
47		vex 2818	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
48	46, 47	xpex 4871	. . . . . . . . 9 ⊢ (𝑥 × 𝑦) ∈ V
49	48	elpw 3680	. . . . . . . 8 ⊢ ((𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌) ↔ (𝑥 × 𝑦) ⊆ (𝑋 × 𝑌))
50	45, 49	sylibr 134	. . . . . . 7 ⊢ ((𝑥 ∈ 𝑅 ∧ 𝑦 ∈ 𝑆) → (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌))
51	50	rgen2 2630	. . . . . 6 ⊢ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌)
52	28	fmpo 6410	. . . . . 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 145	. . . . 5 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)):(𝑅 × 𝑆)⟶𝒫 (𝑋 × 𝑌)
54		frn 5522	. . . . 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 3274	. . 3 ⊢ 𝐵 ⊆ 𝒫 (𝑋 × 𝑌)
57		sspwuni 4081	. . 3 ⊢ (𝐵 ⊆ 𝒫 (𝑋 × 𝑌) ↔ ∪ 𝐵 ⊆ (𝑋 × 𝑌))
58	56, 57	mpbi 145	. 2 ⊢ ∪ 𝐵 ⊆ (𝑋 × 𝑌)
59	39, 58	eqssi 3258	1 ⊢ (𝑋 × 𝑌) = ∪ 𝐵

Syntax hints:   ∧ wa 104   = wceq 1398   ∈ wcel 2205  {cab 2220  ∀wral 2522  ∃wrex 2523   ⊆ wss 3214  𝒫 cpw 3674  ⟨cop 3697  ∪ cuni 3919   × cxp 4752  ran crn 4755  ⟶wf 5353   ∈ cmpo 6060

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348

This theorem is referenced by:  txbasex  15248  txtopon  15253