Theorem txuni2 14979

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 4835	. . 3 ⊢ Rel (𝑋 × 𝑌)
2		txuni2.1	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝑅
3	2	eleq2i 2298	. . . . . . 7 ⊢ (𝑧 ∈ 𝑋 ↔ 𝑧 ∈ ∪ 𝑅)
4		eluni2 3897	. . . . . . 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 2298	. . . . . . 7 ⊢ (𝑤 ∈ 𝑌 ↔ 𝑤 ∈ ∪ 𝑆)
8		eluni2 3897	. . . . . . 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 4755	. . . . 5 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑋 × 𝑌) ↔ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑌))
12		reeanv 2703	. . . . 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 4755	. . . . . 6 ⊢ (⟨𝑧, 𝑤⟩ ∈ (𝑟 × 𝑠) ↔ (𝑧 ∈ 𝑟 ∧ 𝑤 ∈ 𝑠))
15		eqid 2231	. . . . . . . . . 10 ⊢ (𝑟 × 𝑠) = (𝑟 × 𝑠)
16		xpeq1 4739	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑟 → (𝑥 × 𝑦) = (𝑟 × 𝑦))
17	16	eqeq2d 2243	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑟 → ((𝑟 × 𝑠) = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑦)))
18		xpeq2 4740	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑠 → (𝑟 × 𝑦) = (𝑟 × 𝑠))
19	18	eqeq2d 2243	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑠 → ((𝑟 × 𝑠) = (𝑟 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑟 × 𝑠)))
20	17, 19	rspc2ev 2925	. . . . . . . . . 10 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆 ∧ (𝑟 × 𝑠) = (𝑟 × 𝑠)) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
21	15, 20	mp3an3 1362	. . . . . . . . 9 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 (𝑟 × 𝑠) = (𝑥 × 𝑦))
22		vex 2805	. . . . . . . . . . 11 ⊢ 𝑟 ∈ V
23		vex 2805	. . . . . . . . . . 11 ⊢ 𝑠 ∈ V
24	22, 23	xpex 4842	. . . . . . . . . 10 ⊢ (𝑟 × 𝑠) ∈ V
25		eqeq1 2238	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑟 × 𝑠) → (𝑧 = (𝑥 × 𝑦) ↔ (𝑟 × 𝑠) = (𝑥 × 𝑦)))
26	25	2rexbidv 2557	. . . . . . . . . 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 2231	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦))
29	28	rnmpo 6131	. . . . . . . . . . 11 ⊢ ran (𝑥 ∈ 𝑅, 𝑦 ∈ 𝑆 ↦ (𝑥 × 𝑦)) = {𝑧 ∣ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 𝑧 = (𝑥 × 𝑦)}
30	27, 29	eqtri 2252	. . . . . . . . . 10 ⊢ 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝑅 ∃𝑦 ∈ 𝑆 𝑧 = (𝑥 × 𝑦)}
31	24, 26, 30	elab2 2954	. . . . . . . . 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 3921	. . . . . . . 8 ⊢ ((𝑟 × 𝑠) ∈ 𝐵 → (𝑟 × 𝑠) ⊆ ∪ 𝐵)
34	32, 33	syl 14	. . . . . . 7 ⊢ ((𝑟 ∈ 𝑅 ∧ 𝑠 ∈ 𝑆) → (𝑟 × 𝑠) ⊆ ∪ 𝐵)
35	34	sseld 3226	. . . . . 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 2656	. . . 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 4817	. 2 ⊢ (𝑋 × 𝑌) ⊆ ∪ 𝐵
40		elssuni 3921	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝑅 → 𝑥 ⊆ ∪ 𝑅)
41	40, 2	sseqtrrdi 3276	. . . . . . . . 9 ⊢ (𝑥 ∈ 𝑅 → 𝑥 ⊆ 𝑋)
42		elssuni 3921	. . . . . . . . . 10 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ⊆ ∪ 𝑆)
43	42, 6	sseqtrrdi 3276	. . . . . . . . 9 ⊢ (𝑦 ∈ 𝑆 → 𝑦 ⊆ 𝑌)
44		xpss12 4833	. . . . . . . . 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 2805	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
47		vex 2805	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
48	46, 47	xpex 4842	. . . . . . . . 9 ⊢ (𝑥 × 𝑦) ∈ V
49	48	elpw 3658	. . . . . . . 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 2618	. . . . . 6 ⊢ ∀𝑥 ∈ 𝑅 ∀𝑦 ∈ 𝑆 (𝑥 × 𝑦) ∈ 𝒫 (𝑋 × 𝑌)
52	28	fmpo 6365	. . . . . 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 5491	. . . . 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 3259	. . 3 ⊢ 𝐵 ⊆ 𝒫 (𝑋 × 𝑌)
57		sspwuni 4055	. . 3 ⊢ (𝐵 ⊆ 𝒫 (𝑋 × 𝑌) ↔ ∪ 𝐵 ⊆ (𝑋 × 𝑌))
58	56, 57	mpbi 145	. 2 ⊢ ∪ 𝐵 ⊆ (𝑋 × 𝑌)
59	39, 58	eqssi 3243	1 ⊢ (𝑋 × 𝑌) = ∪ 𝐵

Syntax hints:   ∧ wa 104   = wceq 1397   ∈ wcel 2202  {cab 2217  ∀wral 2510  ∃wrex 2511   ⊆ wss 3200  𝒫 cpw 3652  ⟨cop 3672  ∪ cuni 3893   × cxp 4723  ran crn 4726  ⟶wf 5322   ∈ cmpo 6019

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-fv 5334  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303

This theorem is referenced by:  txbasex  14980  txtopon  14985