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Theorem txval 22167
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 3498 . 2 (𝑅𝑉𝑅 ∈ V)
2 elex 3498 . 2 (𝑆𝑊𝑆 ∈ V)
3 mpoeq12 7217 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)))
43rneqd 5796 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)))
5 txval.1 . . . . 5 𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))
64, 5syl6eqr 2877 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = 𝐵)
76fveq2d 6663 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))) = (topGen‘𝐵))
8 df-tx 22165 . . 3 ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
9 fvex 6672 . . 3 (topGen‘𝐵) ∈ V
107, 8, 9ovmpoa 7295 . 2 ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×t 𝑆) = (topGen‘𝐵))
111, 2, 10syl2an 598 1 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480   × cxp 5541  ran crn 5544  cfv 6344  (class class class)co 7146  cmpo 7148  topGenctg 16709   ×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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pr 5318
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4826  df-br 5054  df-opab 5116  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-iota 6303  df-fun 6346  df-fv 6352  df-ov 7149  df-oprab 7150  df-mpo 7151  df-tx 22165
This theorem is referenced by:  eltx  22171  txtop  22172  txtopon  22194  txopn  22205  txss12  22208  txbasval  22209  txcnp  22223  txcnmpt  22227  txrest  22234  txlm  22251  tx2ndc  22254  txflf  22609  mbfimaopnlem  24257
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