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Theorem ptval 21183
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 15928 . . 3 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))}))
21a1i 11 . 2 ((𝐴𝑉𝐹 Fn 𝐴) → ∏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))})))
3 simpr 476 . . . . . . . . . . 11 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
43dmeqd 5248 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝑓 = dom 𝐹)
5 fndm 5904 . . . . . . . . . . 11 (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
65ad2antlr 759 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝐹 = 𝐴)
74, 6eqtrd 2644 . . . . . . . . 9 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝑓 = 𝐴)
87fneq2d 5896 . . . . . . . 8 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑔 Fn dom 𝑓𝑔 Fn 𝐴))
93fveq1d 6105 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑓𝑦) = (𝐹𝑦))
109eleq2d 2673 . . . . . . . . 9 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔𝑦) ∈ (𝑓𝑦) ↔ (𝑔𝑦) ∈ (𝐹𝑦)))
117, 10raleqbidv 3129 . . . . . . . 8 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ dom 𝑓(𝑔𝑦) ∈ (𝑓𝑦) ↔ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦)))
127difeq1d 3689 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (dom 𝑓𝑧) = (𝐴𝑧))
139unieqd 4382 . . . . . . . . . . 11 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑓𝑦) = (𝐹𝑦))
1413eqeq2d 2620 . . . . . . . . . 10 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔𝑦) = (𝑓𝑦) ↔ (𝑔𝑦) = (𝐹𝑦)))
1512, 14raleqbidv 3129 . . . . . . . . 9 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦) ↔ ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)))
1615rexbidv 3034 . . . . . . . 8 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓𝑧)(𝑔𝑦) = (𝑓𝑦) ↔ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)))
178, 11, 163anbi123d 1391 . . . . . . 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))))
187ixpeq1d 7806 . . . . . . . 8 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → X𝑦 ∈ dom 𝑓(𝑔𝑦) = X𝑦𝐴 (𝑔𝑦))
1918eqeq2d 2620 . . . . . . 7 (((𝐴𝑉𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑥 = X𝑦 ∈ dom 𝑓(𝑔𝑦) ↔ 𝑥 = X𝑦𝐴 (𝑔𝑦)))
2017, 19anbi12d 743 . . . . . 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))))
2120exbidv 1837 . . . . 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))))
2221abbidv 2728 . . . 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))})
23 ptval.1 . . . 4 𝐵 = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦𝐴 (𝑔𝑦) ∈ (𝐹𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴𝑧)(𝑔𝑦) = (𝐹𝑦)) ∧ 𝑥 = X𝑦𝐴 (𝑔𝑦))}
2422, 23syl6eqr 2662 . . 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)
2524fveq2d 6107 . 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))
26 fnex 6386 . . 3 ((𝐹 Fn 𝐴𝐴𝑉) → 𝐹 ∈ V)
2726ancoms 468 . 2 ((𝐴𝑉𝐹 Fn 𝐴) → 𝐹 ∈ V)
28 fvex 6113 . . 3 (topGen‘𝐵) ∈ V
2928a1i 11 . 2 ((𝐴𝑉𝐹 Fn 𝐴) → (topGen‘𝐵) ∈ V)
302, 25, 27, 29fvmptd 6197 1 ((𝐴𝑉𝐹 Fn 𝐴) → (∏t𝐹) = (topGen‘𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wex 1695  wcel 1977  {cab 2596  wral 2896  wrex 2897  Vcvv 3173  cdif 3537   cuni 4372  cmpt 4643  dom cdm 5038   Fn wfn 5799  cfv 5804  Xcixp 7794  Fincfn 7841  topGenctg 15921  tcpt 15922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ixp 7795  df-pt 15928
This theorem is referenced by:  pttop  21195  ptopn  21196  ptuni  21207  ptval2  21214  ptpjcn  21224  ptpjopn  21225  ptclsg  21228  ptcnp  21235  prdstopn  21241  xkoptsub  21267  ptcmplem1  21666  tmdgsum2  21710  prdsxmslem2  22144  ptrecube  32579
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