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Theorem ptval 23065
Description: The value of the product topology function. (Contributed by Mario Carneiro, 3-Feb-2015.)
Hypothesis
Ref Expression
ptval.1 𝐡 = {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐴 (π‘”β€˜π‘¦))}
Assertion
Ref Expression
ptval ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) β†’ (∏tβ€˜πΉ) = (topGenβ€˜π΅))
Distinct variable groups:   π‘₯,𝑔,𝑦,𝑧,𝐴   𝑔,𝐹,π‘₯,𝑦,𝑧   𝑔,𝑉,π‘₯,𝑦,𝑧
Allowed substitution hints:   𝐡(π‘₯,𝑦,𝑧,𝑔)

Proof of Theorem ptval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 df-pt 17386 . 2 ∏t = (𝑓 ∈ V ↦ (topGenβ€˜{π‘₯ ∣ βˆƒπ‘”((𝑔 Fn dom 𝑓 ∧ βˆ€π‘¦ ∈ dom 𝑓(π‘”β€˜π‘¦) ∈ (π‘“β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (dom 𝑓 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (π‘“β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ dom 𝑓(π‘”β€˜π‘¦))}))
2 simpr 485 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ 𝑓 = 𝐹)
32dmeqd 5903 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ dom 𝑓 = dom 𝐹)
4 fndm 6649 . . . . . . . . . . 11 (𝐹 Fn 𝐴 β†’ dom 𝐹 = 𝐴)
54ad2antlr 725 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ dom 𝐹 = 𝐴)
63, 5eqtrd 2772 . . . . . . . . 9 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ dom 𝑓 = 𝐴)
76fneq2d 6640 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ (𝑔 Fn dom 𝑓 ↔ 𝑔 Fn 𝐴))
82fveq1d 6890 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ (π‘“β€˜π‘¦) = (πΉβ€˜π‘¦))
98eleq2d 2819 . . . . . . . . 9 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ ((π‘”β€˜π‘¦) ∈ (π‘“β€˜π‘¦) ↔ (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦)))
106, 9raleqbidv 3342 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ (βˆ€π‘¦ ∈ dom 𝑓(π‘”β€˜π‘¦) ∈ (π‘“β€˜π‘¦) ↔ βˆ€π‘¦ ∈ 𝐴 (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦)))
116difeq1d 4120 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ (dom 𝑓 βˆ– 𝑧) = (𝐴 βˆ– 𝑧))
128unieqd 4921 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ βˆͺ (π‘“β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦))
1312eqeq2d 2743 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ ((π‘”β€˜π‘¦) = βˆͺ (π‘“β€˜π‘¦) ↔ (π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)))
1411, 13raleqbidv 3342 . . . . . . . . 9 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ (βˆ€π‘¦ ∈ (dom 𝑓 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (π‘“β€˜π‘¦) ↔ βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)))
1514rexbidv 3178 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ (βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (dom 𝑓 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (π‘“β€˜π‘¦) ↔ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)))
167, 10, 153anbi123d 1436 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ ((𝑔 Fn dom 𝑓 ∧ βˆ€π‘¦ ∈ dom 𝑓(π‘”β€˜π‘¦) ∈ (π‘“β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (dom 𝑓 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (π‘“β€˜π‘¦)) ↔ (𝑔 Fn 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦))))
176ixpeq1d 8899 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ X𝑦 ∈ dom 𝑓(π‘”β€˜π‘¦) = X𝑦 ∈ 𝐴 (π‘”β€˜π‘¦))
1817eqeq2d 2743 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ (π‘₯ = X𝑦 ∈ dom 𝑓(π‘”β€˜π‘¦) ↔ π‘₯ = X𝑦 ∈ 𝐴 (π‘”β€˜π‘¦)))
1916, 18anbi12d 631 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ (((𝑔 Fn dom 𝑓 ∧ βˆ€π‘¦ ∈ dom 𝑓(π‘”β€˜π‘¦) ∈ (π‘“β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (dom 𝑓 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (π‘“β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ dom 𝑓(π‘”β€˜π‘¦)) ↔ ((𝑔 Fn 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐴 (π‘”β€˜π‘¦))))
2019exbidv 1924 . . . . 5 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ (βˆƒπ‘”((𝑔 Fn dom 𝑓 ∧ βˆ€π‘¦ ∈ dom 𝑓(π‘”β€˜π‘¦) ∈ (π‘“β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (dom 𝑓 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (π‘“β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ dom 𝑓(π‘”β€˜π‘¦)) ↔ βˆƒπ‘”((𝑔 Fn 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐴 (π‘”β€˜π‘¦))))
2120abbidv 2801 . . . 4 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn dom 𝑓 ∧ βˆ€π‘¦ ∈ dom 𝑓(π‘”β€˜π‘¦) ∈ (π‘“β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (dom 𝑓 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (π‘“β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ dom 𝑓(π‘”β€˜π‘¦))} = {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐴 (π‘”β€˜π‘¦))})
22 ptval.1 . . . 4 𝐡 = {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn 𝐴 ∧ βˆ€π‘¦ ∈ 𝐴 (π‘”β€˜π‘¦) ∈ (πΉβ€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (𝐴 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (πΉβ€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ 𝐴 (π‘”β€˜π‘¦))}
2321, 22eqtr4di 2790 . . 3 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ {π‘₯ ∣ βˆƒπ‘”((𝑔 Fn dom 𝑓 ∧ βˆ€π‘¦ ∈ dom 𝑓(π‘”β€˜π‘¦) ∈ (π‘“β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (dom 𝑓 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (π‘“β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ dom 𝑓(π‘”β€˜π‘¦))} = 𝐡)
2423fveq2d 6892 . 2 (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) β†’ (topGenβ€˜{π‘₯ ∣ βˆƒπ‘”((𝑔 Fn dom 𝑓 ∧ βˆ€π‘¦ ∈ dom 𝑓(π‘”β€˜π‘¦) ∈ (π‘“β€˜π‘¦) ∧ βˆƒπ‘§ ∈ Fin βˆ€π‘¦ ∈ (dom 𝑓 βˆ– 𝑧)(π‘”β€˜π‘¦) = βˆͺ (π‘“β€˜π‘¦)) ∧ π‘₯ = X𝑦 ∈ dom 𝑓(π‘”β€˜π‘¦))}) = (topGenβ€˜π΅))
25 fnex 7215 . . 3 ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) β†’ 𝐹 ∈ V)
2625ancoms 459 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) β†’ 𝐹 ∈ V)
27 fvexd 6903 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) β†’ (topGenβ€˜π΅) ∈ V)
281, 24, 26, 27fvmptd2 7003 1 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) β†’ (∏tβ€˜πΉ) = (topGenβ€˜π΅))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   ∧ w3a 1087   = wceq 1541  βˆƒwex 1781   ∈ wcel 2106  {cab 2709  βˆ€wral 3061  βˆƒwrex 3070  Vcvv 3474   βˆ– cdif 3944  βˆͺ cuni 4907  dom cdm 5675   Fn wfn 6535  β€˜cfv 6540  Xcixp 8887  Fincfn 8935  topGenctg 17379  βˆtcpt 17380
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ixp 8888  df-pt 17386
This theorem is referenced by:  pttop  23077  ptopn  23078  ptuni  23089  ptval2  23096  ptpjcn  23106  ptpjopn  23107  ptclsg  23110  ptcnp  23117  prdstopn  23123  xkoptsub  23149  ptcmplem1  23547  tmdgsum2  23591  prdsxmslem2  24029  ptrecube  36476
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