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Theorem txunii 22196
Description: The underlying set of the product of two topologies. (Contributed by Jeff Madsen, 15-Jun-2010.)
Hypotheses
Ref Expression
txunii.1 𝑅 ∈ Top
txunii.2 𝑆 ∈ Top
txunii.3 𝑋 = 𝑅
txunii.4 𝑌 = 𝑆
Assertion
Ref Expression
txunii (𝑋 × 𝑌) = (𝑅 ×t 𝑆)

Proof of Theorem txunii
StepHypRef Expression
1 txunii.1 . 2 𝑅 ∈ Top
2 txunii.2 . 2 𝑆 ∈ Top
3 txunii.3 . . 3 𝑋 = 𝑅
4 txunii.4 . . 3 𝑌 = 𝑆
53, 4txuni 22195 . 2 ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑋 × 𝑌) = (𝑅 ×t 𝑆))
61, 2, 5mp2an 691 1 (𝑋 × 𝑌) = (𝑅 ×t 𝑆)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2114   cuni 4813   × cxp 5530  (class class class)co 7140  Topctop 21496   ×t ctx 22163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-topgen 16708  df-top 21497  df-topon 21514  df-bases 21549  df-tx 22165
This theorem is referenced by:  txindis  22237  cxpcn3  25335  tpr2rico  31229  raddcn  31246  sxbrsigalem3  31604  dya2iocucvr  31616  sxbrsigalem1  31617  txsconnlem  32561  cvmlift2lem7  32630  cvmlift2lem9  32632  cvmlift2lem10  32633  cvmlift2lem12  32635  cvmlift2lem13  32636  cvmliftphtlem  32638
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