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Theorem txtopon 12461
Description: The underlying set of the product of two topologies. (Contributed by Mario Carneiro, 22-Aug-2015.) (Revised by Mario Carneiro, 2-Sep-2015.)
Assertion
Ref Expression
txtopon ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))

Proof of Theorem txtopon
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 topontop 12211 . . 3 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
2 topontop 12211 . . 3 (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
3 txtop 12459 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 287 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ Top)
5 eqid 2140 . . . . 5 ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
6 eqid 2140 . . . . 5 𝑅 = 𝑅
7 eqid 2140 . . . . 5 𝑆 = 𝑆
85, 6, 7txuni2 12455 . . . 4 ( 𝑅 × 𝑆) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
9 toponuni 12212 . . . . 5 (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = 𝑅)
10 toponuni 12212 . . . . 5 (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = 𝑆)
11 xpeq12 4562 . . . . 5 ((𝑋 = 𝑅𝑌 = 𝑆) → (𝑋 × 𝑌) = ( 𝑅 × 𝑆))
129, 10, 11syl2an 287 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ( 𝑅 × 𝑆))
135txbasex 12456 . . . . 5 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
14 unitg 12261 . . . . 5 (ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ V → (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)))
1513, 14syl 14 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)))
168, 12, 153eqtr4a 2199 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
175txval 12454 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
1817unieqd 3751 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
1916, 18eqtr4d 2176 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
20 istopon 12210 . 2 ((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ (𝑋 × 𝑌) = (𝑅 ×t 𝑆)))
214, 19, 20sylanbrc 414 1 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wcel 1481  Vcvv 2687   cuni 3740   × cxp 4541  ran crn 4544  cfv 5127  (class class class)co 5778  cmpo 5780  topGenctg 12165  Topctop 12194  TopOnctopon 12207   ×t ctx 12451
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-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4047  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2689  df-sbc 2911  df-csb 3005  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-iun 3819  df-br 3934  df-opab 3994  df-mpt 3995  df-id 4219  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-ov 5781  df-oprab 5782  df-mpo 5783  df-1st 6042  df-2nd 6043  df-topgen 12171  df-top 12195  df-topon 12208  df-bases 12240  df-tx 12452
This theorem is referenced by:  txuni  12462  tx1cn  12468  tx2cn  12469  txcnp  12470  txcnmpt  12472  txdis1cn  12477  txlm  12478  lmcn2  12479  cnmpt12  12486  cnmpt2c  12489  cnmpt21  12490  cnmpt2t  12492  cnmpt22  12493  cnmpt22f  12494  cnmpt2res  12496  cnmptcom  12497  txmetcn  12718  limccnp2lem  12844  limccnp2cntop  12845  dvcnp2cntop  12862  dvaddxxbr  12864  dvmulxxbr  12865  dvcoapbr  12870
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