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Theorem ptval2 23518
Description: The value of the product topology function. (Contributed by Mario Carneiro, 7-Feb-2015.)
Hypotheses
Ref Expression
ptval2.1 𝐽 = (∏tβ€˜πΉ)
ptval2.2 𝑋 = βˆͺ 𝐽
ptval2.3 𝐺 = (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒))
Assertion
Ref Expression
ptval2 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) β†’ 𝐽 = (topGenβ€˜(fiβ€˜({𝑋} βˆͺ ran 𝐺))))
Distinct variable groups:   𝑒,π‘˜,𝑀,𝐴   π‘˜,𝐹,𝑒,𝑀   π‘˜,𝑉,𝑒,𝑀   𝑀,𝑋
Allowed substitution hints:   𝐺(𝑀,𝑒,π‘˜)   𝐽(𝑀,𝑒,π‘˜)   𝑋(𝑒,π‘˜)

Proof of Theorem ptval2
Dummy variables 𝑔 𝑛 π‘₯ 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ffn 6722 . . 3 (𝐹:𝐴⟢Top β†’ 𝐹 Fn 𝐴)
2 ptval2.1 . . . 4 𝐽 = (∏tβ€˜πΉ)
3 eqid 2728 . . . . 5 {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‘¦))} = {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‘¦))}
43ptval 23487 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) β†’ (∏tβ€˜πΉ) = (topGenβ€˜{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‘¦))}))
52, 4eqtrid 2780 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) β†’ 𝐽 = (topGenβ€˜{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‘¦))}))
61, 5sylan2 592 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) β†’ 𝐽 = (topGenβ€˜{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‘¦))}))
7 eqid 2728 . . . . 5 X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) = X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›)
83, 7ptbasfi 23498 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) 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‘¦))} = (fiβ€˜({X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›)} βˆͺ ran (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) ↦ (π‘€β€˜π‘˜)) β€œ 𝑒)))))
92ptuni 23511 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) β†’ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) = βˆͺ 𝐽)
10 ptval2.2 . . . . . . . 8 𝑋 = βˆͺ 𝐽
119, 10eqtr4di 2786 . . . . . . 7 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) β†’ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) = 𝑋)
1211sneqd 4641 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) β†’ {X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›)} = {𝑋})
13113ad2ant1 1131 . . . . . . . . . . . 12 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) ∧ π‘˜ ∈ 𝐴 ∧ 𝑒 ∈ (πΉβ€˜π‘˜)) β†’ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) = 𝑋)
1413mpteq1d 5243 . . . . . . . . . . 11 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) ∧ π‘˜ ∈ 𝐴 ∧ 𝑒 ∈ (πΉβ€˜π‘˜)) β†’ (𝑀 ∈ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) ↦ (π‘€β€˜π‘˜)) = (𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)))
1514cnveqd 5878 . . . . . . . . . 10 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) ∧ π‘˜ ∈ 𝐴 ∧ 𝑒 ∈ (πΉβ€˜π‘˜)) β†’ β—‘(𝑀 ∈ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) ↦ (π‘€β€˜π‘˜)) = β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)))
1615imaeq1d 6062 . . . . . . . . 9 (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) ∧ π‘˜ ∈ 𝐴 ∧ 𝑒 ∈ (πΉβ€˜π‘˜)) β†’ (β—‘(𝑀 ∈ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) ↦ (π‘€β€˜π‘˜)) β€œ 𝑒) = (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒))
1716mpoeq3dva 7497 . . . . . . . 8 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) β†’ (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) ↦ (π‘€β€˜π‘˜)) β€œ 𝑒)) = (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒)))
18 ptval2.3 . . . . . . . 8 𝐺 = (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ 𝑋 ↦ (π‘€β€˜π‘˜)) β€œ 𝑒))
1917, 18eqtr4di 2786 . . . . . . 7 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) β†’ (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) ↦ (π‘€β€˜π‘˜)) β€œ 𝑒)) = 𝐺)
2019rneqd 5940 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) β†’ ran (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) ↦ (π‘€β€˜π‘˜)) β€œ 𝑒)) = ran 𝐺)
2112, 20uneq12d 4163 . . . . 5 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) β†’ ({X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›)} βˆͺ ran (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) ↦ (π‘€β€˜π‘˜)) β€œ 𝑒))) = ({𝑋} βˆͺ ran 𝐺))
2221fveq2d 6901 . . . 4 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) β†’ (fiβ€˜({X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›)} βˆͺ ran (π‘˜ ∈ 𝐴, 𝑒 ∈ (πΉβ€˜π‘˜) ↦ (β—‘(𝑀 ∈ X𝑛 ∈ 𝐴 βˆͺ (πΉβ€˜π‘›) ↦ (π‘€β€˜π‘˜)) β€œ 𝑒)))) = (fiβ€˜({𝑋} βˆͺ ran 𝐺)))
238, 22eqtrd 2768 . . 3 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) 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‘¦))} = (fiβ€˜({𝑋} βˆͺ ran 𝐺)))
2423fveq2d 6901 . 2 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) β†’ (topGenβ€˜{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‘¦))}) = (topGenβ€˜(fiβ€˜({𝑋} βˆͺ ran 𝐺))))
256, 24eqtrd 2768 1 ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟢Top) β†’ 𝐽 = (topGenβ€˜(fiβ€˜({𝑋} βˆͺ ran 𝐺))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1534  βˆƒwex 1774   ∈ wcel 2099  {cab 2705  βˆ€wral 3058  βˆƒwrex 3067   βˆ– cdif 3944   βˆͺ cun 3945  {csn 4629  βˆͺ cuni 4908   ↦ cmpt 5231  β—‘ccnv 5677  ran crn 5679   β€œ cima 5681   Fn wfn 6543  βŸΆwf 6544  β€˜cfv 6548   ∈ cmpo 7422  Xcixp 8916  Fincfn 8964  ficfi 9434  topGenctg 17419  βˆtcpt 17420  Topctop 22808
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-1o 8487  df-er 8725  df-ixp 8917  df-en 8965  df-dom 8966  df-fin 8968  df-fi 9435  df-topgen 17425  df-pt 17426  df-top 22809  df-bases 22862
This theorem is referenced by:  ptrescn  23556  ptrest  37092
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