MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  txval Structured version   Visualization version   GIF version

Theorem txval 21589
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 3364 . 2 (𝑅𝑉𝑅 ∈ V)
2 elex 3364 . 2 (𝑆𝑊𝑆 ∈ V)
3 mpt2eq12 6863 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)))
43rneqd 5492 . . . . 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 2823 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = 𝐵)
76fveq2d 6337 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))) = (topGen‘𝐵))
8 df-tx 21587 . . 3 ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
9 fvex 6343 . . 3 (topGen‘𝐵) ∈ V
107, 8, 9ovmpt2a 6939 . 2 ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×t 𝑆) = (topGen‘𝐵))
111, 2, 10syl2an 577 1 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  Vcvv 3351   × cxp 5248  ran crn 5251  cfv 6032  (class class class)co 6794  cmpt2 6796  topGenctg 16307   ×t ctx 21585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-iota 5995  df-fun 6034  df-fv 6040  df-ov 6797  df-oprab 6798  df-mpt2 6799  df-tx 21587
This theorem is referenced by:  eltx  21593  txtop  21594  txtopon  21616  txopn  21627  txss12  21630  txbasval  21631  txcnp  21645  txcnmpt  21649  txrest  21656  txlm  21673  tx2ndc  21676  txflf  22031  mbfimaopnlem  23643
  Copyright terms: Public domain W3C validator