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Theorem ptval 23593
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 17490 . 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 5918 . . . . . . . . . 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 2774 . . . . . . . . 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 2824 . . . . . . . . 9 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔𝑦) ∈ (𝑓𝑦) ↔ (𝑔𝑦) ∈ (𝐹𝑦)))
106, 9raleqbidv 3343 . . . . . . . 8 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ↔ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦)))
116difeq1d 4134 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (dom 𝑓𝑧) = (𝐴𝑧))
128unieqd 4924 . . . . . . . . . . 11 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑓𝑦) = (𝐹𝑦))
1312eqeq2d 2745 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔𝑦) = (𝑓𝑦) ↔ (𝑔𝑦) = (𝐹𝑦)))
1411, 13raleqbidv 3343 . . . . . . . . 9 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦) ↔ ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)))
1514rexbidv 3176 . . . . . . . 8 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦) ↔ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)))
167, 10, 153anbi123d 1435 . . . . . . 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 8947 . . . . . . . 8 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → X𝑦 ∈ dom 𝑓(𝑔𝑦) = X𝑦𝐴 (𝑔𝑦))
1817eqeq2d 2745 . . . . . . 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 1918 . . . . 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 2805 . . . 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 2792 . . 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 7236 . . 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 7023 1 ((𝐴𝑉𝐹 Fn 𝐴) → (∏t𝐹) = (topGen‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1536  wex 1775  wcel 2105  {cab 2711  wral 3058  wrex 3067  Vcvv 3477  cdif 3959   cuni 4911  dom cdm 5688   Fn wfn 6557  cfv 6562  Xcixp 8935  Fincfn 8983  topGenctg 17483  tcpt 17484
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ixp 8936  df-pt 17490
This theorem is referenced by:  pttop  23605  ptopn  23606  ptuni  23617  ptval2  23624  ptpjcn  23634  ptpjopn  23635  ptclsg  23638  ptcnp  23645  prdstopn  23651  xkoptsub  23677  ptcmplem1  24075  tmdgsum2  24119  prdsxmslem2  24557  ptrecube  37606
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