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Theorem txval 12602
 Description: Value of the binary topological product operation. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 30-Aug-2015.)
Hypothesis
Ref Expression
txval.1 𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))
Assertion
Ref Expression
txval ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵))
Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝑆,𝑦
Allowed substitution hints:   𝐵(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem txval
Dummy variables 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elex 2720 . . . 4 (𝑅𝑉𝑅 ∈ V)
21adantr 274 . . 3 ((𝑅𝑉𝑆𝑊) → 𝑅 ∈ V)
3 elex 2720 . . . 4 (𝑆𝑊𝑆 ∈ V)
43adantl 275 . . 3 ((𝑅𝑉𝑆𝑊) → 𝑆 ∈ V)
5 mpoexga 6150 . . . 4 ((𝑅𝑉𝑆𝑊) → (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) ∈ V)
6 rnexg 4844 . . . 4 ((𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) ∈ V → ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) ∈ V)
7 tgvalex 12397 . . . 4 (ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) ∈ V → (topGen‘ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))) ∈ V)
85, 6, 73syl 17 . . 3 ((𝑅𝑉𝑆𝑊) → (topGen‘ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))) ∈ V)
9 mpoeq12 5871 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)))
109rneqd 4808 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)))
1110fveq2d 5465 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))) = (topGen‘ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))))
12 df-tx 12600 . . . 4 ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
1311, 12ovmpoga 5940 . . 3 ((𝑅 ∈ V ∧ 𝑆 ∈ V ∧ (topGen‘ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))) ∈ V) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))))
142, 4, 8, 13syl3anc 1217 . 2 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))))
15 txval.1 . . 3 𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))
1615fveq2i 5464 . 2 (topGen‘𝐵) = (topGen‘ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)))
1714, 16eqtr4di 2205 1 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 2125  Vcvv 2709   × cxp 4577  ran crn 4580  ‘cfv 5163  (class class class)co 5814   ∈ cmpo 5816  topGenctg 12313   ×t ctx 12599 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 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-ral 2437  df-rex 2438  df-reu 2439  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-ov 5817  df-oprab 5818  df-mpo 5819  df-1st 6078  df-2nd 6079  df-topgen 12319  df-tx 12600 This theorem is referenced by:  eltx  12606  txtop  12607  txtopon  12609  txopn  12612  txss12  12613  txbasval  12614  txcnp  12618  txcnmpt  12620  txrest  12623  txlm  12626  xmettxlem  12856  xmettx  12857
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