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Theorem txval 21696
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 3400 . 2 (𝑅𝑉𝑅 ∈ V)
2 elex 3400 . 2 (𝑆𝑊𝑆 ∈ V)
3 mpt2eq12 6949 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)))
43rneqd 5556 . . . . 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 2851 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = 𝐵)
76fveq2d 6415 . . 3 ((𝑟 = 𝑅𝑠 = 𝑆) → (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))) = (topGen‘𝐵))
8 df-tx 21694 . . 3 ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
9 fvex 6424 . . 3 (topGen‘𝐵) ∈ V
107, 8, 9ovmpt2a 7025 . 2 ((𝑅 ∈ V ∧ 𝑆 ∈ V) → (𝑅 ×t 𝑆) = (topGen‘𝐵))
111, 2, 10syl2an 590 1 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3385   × cxp 5310  ran crn 5313  cfv 6101  (class class class)co 6878  cmpt2 6880  topGenctg 16413   ×t ctx 21692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-tx 21694
This theorem is referenced by:  eltx  21700  txtop  21701  txtopon  21723  txopn  21734  txss12  21737  txbasval  21738  txcnp  21752  txcnmpt  21756  txrest  21763  txlm  21780  tx2ndc  21783  txflf  22138  mbfimaopnlem  23763
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