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Theorem ptval 22185
 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 16713 . 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 488 . . . . . . . . . . 11 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
32dmeqd 5739 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝑓 = dom 𝐹)
4 fndm 6426 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
54ad2antlr 726 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝐹 = 𝐴)
63, 5eqtrd 2833 . . . . . . . . 9 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝑓 = 𝐴)
76fneq2d 6418 . . . . . . . 8 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑔 Fn dom 𝑓𝑔 Fn 𝐴))
82fveq1d 6648 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑓𝑦) = (𝐹𝑦))
98eleq2d 2875 . . . . . . . . 9 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔𝑦) ∈ (𝑓𝑦) ↔ (𝑔𝑦) ∈ (𝐹𝑦)))
106, 9raleqbidv 3354 . . . . . . . 8 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ↔ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦)))
116difeq1d 4049 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (dom 𝑓𝑧) = (𝐴𝑧))
128unieqd 4815 . . . . . . . . . . 11 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑓𝑦) = (𝐹𝑦))
1312eqeq2d 2809 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔𝑦) = (𝑓𝑦) ↔ (𝑔𝑦) = (𝐹𝑦)))
1411, 13raleqbidv 3354 . . . . . . . . 9 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦) ↔ ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)))
1514rexbidv 3256 . . . . . . . 8 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦) ↔ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)))
167, 10, 153anbi123d 1433 . . . . . . 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 8459 . . . . . . . 8 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → X𝑦 ∈ dom 𝑓(𝑔𝑦) = X𝑦𝐴 (𝑔𝑦))
1817eqeq2d 2809 . . . . . . 7 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑥 = X𝑦 ∈ dom 𝑓(𝑔𝑦) ↔ 𝑥 = X𝑦𝐴 (𝑔𝑦)))
1916, 18anbi12d 633 . . . . . 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 1922 . . . . 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 2862 . . . 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 2851 . . 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 6650 . 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 6958 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
2625ancoms 462 . 2 ((𝐴𝑉𝐹 Fn 𝐴) → 𝐹 ∈ V)
27 fvexd 6661 . 2 ((𝐴𝑉𝐹 Fn 𝐴) → (topGen‘𝐵) ∈ V)
281, 24, 26, 27fvmptd2 6754 1 ((𝐴𝑉𝐹 Fn 𝐴) → (∏t𝐹) = (topGen‘𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107  Vcvv 3441   ∖ cdif 3878  ∪ cuni 4801  dom cdm 5520   Fn wfn 6320  ‘cfv 6325  Xcixp 8447  Fincfn 8495  topGenctg 16706  ∏tcpt 16707 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5155  ax-sep 5168  ax-nul 5175  ax-pr 5296 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4884  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5426  df-xp 5526  df-rel 5527  df-cnv 5528  df-co 5529  df-dm 5530  df-rn 5531  df-res 5532  df-ima 5533  df-iota 6284  df-fun 6327  df-fn 6328  df-f 6329  df-f1 6330  df-fo 6331  df-f1o 6332  df-fv 6333  df-ixp 8448  df-pt 16713 This theorem is referenced by:  pttop  22197  ptopn  22198  ptuni  22209  ptval2  22216  ptpjcn  22226  ptpjopn  22227  ptclsg  22230  ptcnp  22237  prdstopn  22243  xkoptsub  22269  ptcmplem1  22667  tmdgsum2  22711  prdsxmslem2  23146  ptrecube  35076
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