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Theorem txtopon 13395
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 13145 . . 3 (𝑅 ∈ (TopOn‘𝑋) → 𝑅 ∈ Top)
2 topontop 13145 . . 3 (𝑆 ∈ (TopOn‘𝑌) → 𝑆 ∈ Top)
3 txtop 13393 . . 3 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑅 ×t 𝑆) ∈ Top)
41, 2, 3syl2an 289 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ Top)
5 eqid 2177 . . . . 5 ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
6 eqid 2177 . . . . 5 𝑅 = 𝑅
7 eqid 2177 . . . . 5 𝑆 = 𝑆
85, 6, 7txuni2 13389 . . . 4 ( 𝑅 × 𝑆) = ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))
9 toponuni 13146 . . . . 5 (𝑅 ∈ (TopOn‘𝑋) → 𝑋 = 𝑅)
10 toponuni 13146 . . . . 5 (𝑆 ∈ (TopOn‘𝑌) → 𝑌 = 𝑆)
11 xpeq12 4641 . . . . 5 ((𝑋 = 𝑅𝑌 = 𝑆) → (𝑋 × 𝑌) = ( 𝑅 × 𝑆))
129, 10, 11syl2an 289 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = ( 𝑅 × 𝑆))
135txbasex 13390 . . . . 5 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣)) ∈ V)
14 unitg 13195 . . . . 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 2236 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
175txval 13388 . . . 4 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
1817unieqd 3818 . . 3 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑢𝑅, 𝑣𝑆 ↦ (𝑢 × 𝑣))))
1916, 18eqtr4d 2213 . 2 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
20 istopon 13144 . 2 ((𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)) ↔ ((𝑅 ×t 𝑆) ∈ Top ∧ (𝑋 × 𝑌) = (𝑅 ×t 𝑆)))
214, 19, 20sylanbrc 417 1 ((𝑅 ∈ (TopOn‘𝑋) ∧ 𝑆 ∈ (TopOn‘𝑌)) → (𝑅 ×t 𝑆) ∈ (TopOn‘(𝑋 × 𝑌)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2737   cuni 3807   × cxp 4620  ran crn 4623  cfv 5211  (class class class)co 5868  cmpo 5870  topGenctg 12638  Topctop 13128  TopOnctopon 13141   ×t ctx 13385
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-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-topgen 12644  df-top 13129  df-topon 13142  df-bases 13174  df-tx 13386
This theorem is referenced by:  txuni  13396  tx1cn  13402  tx2cn  13403  txcnp  13404  txcnmpt  13406  txdis1cn  13411  txlm  13412  lmcn2  13413  cnmpt12  13420  cnmpt2c  13423  cnmpt21  13424  cnmpt2t  13426  cnmpt22  13427  cnmpt22f  13428  cnmpt2res  13430  cnmptcom  13431  txmetcn  13652  limccnp2lem  13778  limccnp2cntop  13779  dvcnp2cntop  13796  dvaddxxbr  13798  dvmulxxbr  13799  dvcoapbr  13804
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