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Theorem ptval 23578
Description: The value of the product topology function. (Contributed by Mario Carneiro, 3-Feb-2015.)
Hypothesis
Ref Expression
ptval.1 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
Assertion
Ref Expression
ptval ((𝐴𝑉𝐹 Fn 𝐴) → (∏t𝐹) = (topGen‘𝐵))
Distinct variable groups:   𝑥,𝑔,𝑦,𝑧,𝐴   𝑔,𝐹,𝑥,𝑦,𝑧   𝑔,𝑉,𝑥,𝑦,𝑧
Allowed substitution hints:   𝐵(𝑥,𝑦,𝑧,𝑔)

Proof of Theorem ptval
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 df-pt 17489 . 2 t = (𝑓 ∈ V ↦ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔𝑦))}))
2 simpr 484 . . . . . . . . . . 11 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
32dmeqd 5916 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝑓 = dom 𝐹)
4 fndm 6671 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
54ad2antlr 727 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝐹 = 𝐴)
63, 5eqtrd 2777 . . . . . . . . 9 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝑓 = 𝐴)
76fneq2d 6662 . . . . . . . 8 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑔 Fn dom 𝑓𝑔 Fn 𝐴))
82fveq1d 6908 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑓𝑦) = (𝐹𝑦))
98eleq2d 2827 . . . . . . . . 9 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔𝑦) ∈ (𝑓𝑦) ↔ (𝑔𝑦) ∈ (𝐹𝑦)))
106, 9raleqbidv 3346 . . . . . . . 8 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ↔ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦)))
116difeq1d 4125 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (dom 𝑓𝑧) = (𝐴𝑧))
128unieqd 4920 . . . . . . . . . . 11 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑓𝑦) = (𝐹𝑦))
1312eqeq2d 2748 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔𝑦) = (𝑓𝑦) ↔ (𝑔𝑦) = (𝐹𝑦)))
1411, 13raleqbidv 3346 . . . . . . . . 9 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦) ↔ ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)))
1514rexbidv 3179 . . . . . . . 8 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦) ↔ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)))
167, 10, 153anbi123d 1438 . . . . . . 7 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦)) ↔ (𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦))))
176ixpeq1d 8949 . . . . . . . 8 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → X𝑦 ∈ dom 𝑓(𝑔𝑦) = X𝑦𝐴 (𝑔𝑦))
1817eqeq2d 2748 . . . . . . 7 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑥 = X𝑦 ∈ dom 𝑓(𝑔𝑦) ↔ 𝑥 = X𝑦𝐴 (𝑔𝑦)))
1916, 18anbi12d 632 . . . . . 6 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔𝑦)) ↔ ((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))))
2019exbidv 1921 . . . . 5 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔𝑦)) ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))))
2120abbidv 2808 . . . 4 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → {𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔𝑦))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))})
22 ptval.1 . . . 4 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
2321, 22eqtr4di 2795 . . 3 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → {𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔𝑦))} = 𝐵)
2423fveq2d 6910 . 2 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔𝑦))}) = (topGen‘𝐵))
25 fnex 7237 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
2625ancoms 458 . 2 ((𝐴𝑉𝐹 Fn 𝐴) → 𝐹 ∈ V)
27 fvexd 6921 . 2 ((𝐴𝑉𝐹 Fn 𝐴) → (topGen‘𝐵) ∈ V)
281, 24, 26, 27fvmptd2 7024 1 ((𝐴𝑉𝐹 Fn 𝐴) → (∏t𝐹) = (topGen‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1087   = wceq 1540  wex 1779  wcel 2108  {cab 2714  wral 3061  wrex 3070  Vcvv 3480  cdif 3948   cuni 4907  dom cdm 5685   Fn wfn 6556  cfv 6561  Xcixp 8937  Fincfn 8985  topGenctg 17482  tcpt 17483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ixp 8938  df-pt 17489
This theorem is referenced by:  pttop  23590  ptopn  23591  ptuni  23602  ptval2  23609  ptpjcn  23619  ptpjopn  23620  ptclsg  23623  ptcnp  23630  prdstopn  23636  xkoptsub  23662  ptcmplem1  24060  tmdgsum2  24104  prdsxmslem2  24542  ptrecube  37627
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