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Theorem txval 13388
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 2748 . . . 4 (𝑅𝑉𝑅 ∈ V)
21adantr 276 . . 3 ((𝑅𝑉𝑆𝑊) → 𝑅 ∈ V)
3 elex 2748 . . . 4 (𝑆𝑊𝑆 ∈ V)
43adantl 277 . . 3 ((𝑅𝑉𝑆𝑊) → 𝑆 ∈ V)
5 mpoexga 6206 . . . 4 ((𝑅𝑉𝑆𝑊) → (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) ∈ V)
6 rnexg 4887 . . . 4 ((𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) ∈ V → ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)) ∈ V)
7 tgvalex 13183 . . . 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 5928 . . . . . 6 ((𝑟 = 𝑅𝑠 = 𝑆) → (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)))
109rneqd 4851 . . . . 5 ((𝑟 = 𝑅𝑠 = 𝑆) → ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦)) = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)))
1110fveq2d 5514 . . . 4 ((𝑟 = 𝑅𝑠 = 𝑆) → (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))) = (topGen‘ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))))
12 df-tx 13386 . . . 4 ×t = (𝑟 ∈ V, 𝑠 ∈ V ↦ (topGen‘ran (𝑥𝑟, 𝑦𝑠 ↦ (𝑥 × 𝑦))))
1311, 12ovmpoga 5997 . . 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 1238 . 2 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))))
15 txval.1 . . 3 𝐵 = ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦))
1615fveq2i 5513 . 2 (topGen‘𝐵) = (topGen‘ran (𝑥𝑅, 𝑦𝑆 ↦ (𝑥 × 𝑦)))
1714, 16eqtr4di 2228 1 ((𝑅𝑉𝑆𝑊) → (𝑅 ×t 𝑆) = (topGen‘𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148  Vcvv 2737   × cxp 4620  ran crn 4623  cfv 5211  (class class class)co 5868  cmpo 5870  topGenctg 12638   ×t ctx 13385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-f1 5216  df-fo 5217  df-f1o 5218  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-topgen 12644  df-tx 13386
This theorem is referenced by:  eltx  13392  txtop  13393  txtopon  13395  txopn  13398  txss12  13399  txbasval  13400  txcnp  13404  txcnmpt  13406  txrest  13409  txlm  13412  xmettxlem  13642  xmettx  13643
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