Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfac21 Structured version   Visualization version   GIF version
Theorem dfac21 42110
Description: Tychonoff's theorem is a choice equivalent. Definition AC21 of Schechter p. 461. (Contributed by Stefan O'Rear, 22-Feb-2015.) (Revised by Mario Carneiro, 27-Aug-2015.)
Assertion
Ref Expression
dfac21 (CHOICE ↔ βˆ€π‘“(𝑓:dom π‘“βŸΆComp β†’ (∏tβ€˜π‘“) ∈ Comp))
Proof of Theorem dfac21
Dummy variables 𝑔 𝑦 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3478 . . . . . . 7 𝑓 ∈ V
21dmex 7904 . . . . . 6 dom 𝑓 ∈ V
32a1i 11 . . . . 5 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ dom 𝑓 ∈ V)
4 simpr 485 . . . . 5 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ 𝑓:dom π‘“βŸΆComp)
5 fvex 6904 . . . . . . . 8 (∏tβ€˜π‘“) ∈ V
65uniex 7733 . . . . . . 7 βˆͺ (∏tβ€˜π‘“) ∈ V
7 acufl 23641 . . . . . . . 8 (CHOICE β†’ UFL = V)
87adantr 481 . . . . . . 7 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ UFL = V)
96, 8eleqtrrid 2840 . . . . . 6 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ βˆͺ (∏tβ€˜π‘“) ∈ UFL)
10 simpl 483 . . . . . . . 8 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ CHOICE)
11 dfac10 10134 . . . . . . . 8 (CHOICE ↔ dom card = V)
1210, 11sylib 217 . . . . . . 7 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ dom card = V)
136, 12eleqtrrid 2840 . . . . . 6 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ βˆͺ (∏tβ€˜π‘“) ∈ dom card)
149, 13elind 4194 . . . . 5 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ βˆͺ (∏tβ€˜π‘“) ∈ (UFL ∩ dom card))
15 eqid 2732 . . . . . 6 (∏tβ€˜π‘“) = (∏tβ€˜π‘“)
16 eqid 2732 . . . . . 6 βˆͺ (∏tβ€˜π‘“) = βˆͺ (∏tβ€˜π‘“)
1715, 16ptcmpg 23781 . . . . 5 ((dom 𝑓 ∈ V ∧ 𝑓:dom π‘“βŸΆComp ∧ βˆͺ (∏tβ€˜π‘“) ∈ (UFL ∩ dom card)) β†’ (∏tβ€˜π‘“) ∈ Comp)
183, 4, 14, 17syl3anc 1371 . . . 4 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ (∏tβ€˜π‘“) ∈ Comp)
1918ex 413 . . 3 (CHOICE β†’ (𝑓:dom π‘“βŸΆComp β†’ (∏tβ€˜π‘“) ∈ Comp))
2019alrimiv 1930 . 2 (CHOICE β†’ βˆ€π‘“(𝑓:dom π‘“βŸΆComp β†’ (∏tβ€˜π‘“) ∈ Comp))
21 fvex 6904 . . . . . . . . 9 (π‘”β€˜π‘¦) ∈ V
22 kelac2lem 42108 . . . . . . . . 9 ((π‘”β€˜π‘¦) ∈ V β†’ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}) ∈ Comp)
2321, 22mp1i 13 . . . . . . . 8 (((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) ∧ 𝑦 ∈ dom 𝑔) β†’ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}) ∈ Comp)
2423fmpttd 7116 . . . . . . 7 ((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) β†’ (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})):dom π‘”βŸΆComp)
2524ffdmd 6748 . . . . . 6 ((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) β†’ (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))⟢Comp)
26 vex 3478 . . . . . . . . 9 𝑔 ∈ V
2726dmex 7904 . . . . . . . 8 dom 𝑔 ∈ V
2827mptex 7227 . . . . . . 7 (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})) ∈ V
29 id 22 . . . . . . . . 9 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})) β†’ 𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})))
30 dmeq 5903 . . . . . . . . 9 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})) β†’ dom 𝑓 = dom (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})))
3129, 30feq12d 6705 . . . . . . . 8 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})) β†’ (𝑓:dom π‘“βŸΆComp ↔ (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))⟢Comp))
32 fveq2 6891 . . . . . . . . 9 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})) β†’ (∏tβ€˜π‘“) = (∏tβ€˜(𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))))
3332eleq1d 2818 . . . . . . . 8 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})) β†’ ((∏tβ€˜π‘“) ∈ Comp ↔ (∏tβ€˜(𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))) ∈ Comp))
3431, 33imbi12d 344 . . . . . . 7 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})) β†’ ((𝑓:dom π‘“βŸΆComp β†’ (∏tβ€˜π‘“) ∈ Comp) ↔ ((𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))⟢Comp β†’ (∏tβ€˜(𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))) ∈ Comp)))
3528, 34spcv 3595 . . . . . 6 (βˆ€π‘“(𝑓:dom π‘“βŸΆComp β†’ (∏tβ€˜π‘“) ∈ Comp) β†’ ((𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))⟢Comp β†’ (∏tβ€˜(𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))) ∈ Comp))
3625, 35syl5com 31 . . . . 5 ((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) β†’ (βˆ€π‘“(𝑓:dom π‘“βŸΆComp β†’ (∏tβ€˜π‘“) ∈ Comp) β†’ (∏tβ€˜(𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))) ∈ Comp))
37 fvex 6904 . . . . . . . 8 (π‘”β€˜π‘₯) ∈ V
3837a1i 11 . . . . . . 7 ((((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) ∧ (∏tβ€˜(𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))) ∈ Comp) ∧ π‘₯ ∈ dom 𝑔) β†’ (π‘”β€˜π‘₯) ∈ V)
39 df-nel 3047 . . . . . . . . . . 11 (βˆ… βˆ‰ ran 𝑔 ↔ Β¬ βˆ… ∈ ran 𝑔)
4039biimpi 215 . . . . . . . . . 10 (βˆ… βˆ‰ ran 𝑔 β†’ Β¬ βˆ… ∈ ran 𝑔)
4140ad2antlr 725 . . . . . . . . 9 (((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) ∧ π‘₯ ∈ dom 𝑔) β†’ Β¬ βˆ… ∈ ran 𝑔)
42 fvelrn 7078 . . . . . . . . . . . 12 ((Fun 𝑔 ∧ π‘₯ ∈ dom 𝑔) β†’ (π‘”β€˜π‘₯) ∈ ran 𝑔)
4342adantlr 713 . . . . . . . . . . 11 (((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) ∧ π‘₯ ∈ dom 𝑔) β†’ (π‘”β€˜π‘₯) ∈ ran 𝑔)
44 eleq1 2821 . . . . . . . . . . 11 ((π‘”β€˜π‘₯) = βˆ… β†’ ((π‘”β€˜π‘₯) ∈ ran 𝑔 ↔ βˆ… ∈ ran 𝑔))
4543, 44syl5ibcom 244 . . . . . . . . . 10 (((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) ∧ π‘₯ ∈ dom 𝑔) β†’ ((π‘”β€˜π‘₯) = βˆ… β†’ βˆ… ∈ ran 𝑔))
4645necon3bd 2954 . . . . . . . . 9 (((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) ∧ π‘₯ ∈ dom 𝑔) β†’ (Β¬ βˆ… ∈ ran 𝑔 β†’ (π‘”β€˜π‘₯) β‰  βˆ…))
4741, 46mpd 15 . . . . . . . 8 (((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) ∧ π‘₯ ∈ dom 𝑔) β†’ (π‘”β€˜π‘₯) β‰  βˆ…)
4847adantlr 713 . . . . . . 7 ((((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) ∧ (∏tβ€˜(𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))) ∈ Comp) ∧ π‘₯ ∈ dom 𝑔) β†’ (π‘”β€˜π‘₯) β‰  βˆ…)
49 fveq2 6891 . . . . . . . . . . . . . 14 (𝑦 = π‘₯ β†’ (π‘”β€˜π‘¦) = (π‘”β€˜π‘₯))
5049unieqd 4922 . . . . . . . . . . . . . . . 16 (𝑦 = π‘₯ β†’ βˆͺ (π‘”β€˜π‘¦) = βˆͺ (π‘”β€˜π‘₯))
5150pweqd 4619 . . . . . . . . . . . . . . 15 (𝑦 = π‘₯ β†’ 𝒫 βˆͺ (π‘”β€˜π‘¦) = 𝒫 βˆͺ (π‘”β€˜π‘₯))
5251sneqd 4640 . . . . . . . . . . . . . 14 (𝑦 = π‘₯ β†’ {𝒫 βˆͺ (π‘”β€˜π‘¦)} = {𝒫 βˆͺ (π‘”β€˜π‘₯)})
5349, 52preq12d 4745 . . . . . . . . . . . . 13 (𝑦 = π‘₯ β†’ {(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}} = {(π‘”β€˜π‘₯), {𝒫 βˆͺ (π‘”β€˜π‘₯)}})
5453fveq2d 6895 . . . . . . . . . . . 12 (𝑦 = π‘₯ β†’ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}) = (topGenβ€˜{(π‘”β€˜π‘₯), {𝒫 βˆͺ (π‘”β€˜π‘₯)}}))
5554cbvmptv 5261 . . . . . . . . . . 11 (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})) = (π‘₯ ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘₯), {𝒫 βˆͺ (π‘”β€˜π‘₯)}}))
5655fveq2i 6894 . . . . . . . . . 10 (∏tβ€˜(𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))) = (∏tβ€˜(π‘₯ ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘₯), {𝒫 βˆͺ (π‘”β€˜π‘₯)}})))
5756eleq1i 2824 . . . . . . . . 9 ((∏tβ€˜(𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))) ∈ Comp ↔ (∏tβ€˜(π‘₯ ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘₯), {𝒫 βˆͺ (π‘”β€˜π‘₯)}}))) ∈ Comp)
5857biimpi 215 . . . . . . . 8 ((∏tβ€˜(𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))) ∈ Comp β†’ (∏tβ€˜(π‘₯ ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘₯), {𝒫 βˆͺ (π‘”β€˜π‘₯)}}))) ∈ Comp)
5958adantl 482 . . . . . . 7 (((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) ∧ (∏tβ€˜(𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))) ∈ Comp) β†’ (∏tβ€˜(π‘₯ ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘₯), {𝒫 βˆͺ (π‘”β€˜π‘₯)}}))) ∈ Comp)
6038, 48, 59kelac2 42109 . . . . . 6 (((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) ∧ (∏tβ€˜(𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))) ∈ Comp) β†’ Xπ‘₯ ∈ dom 𝑔(π‘”β€˜π‘₯) β‰  βˆ…)
6160ex 413 . . . . 5 ((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) β†’ ((∏tβ€˜(𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}))) ∈ Comp β†’ Xπ‘₯ ∈ dom 𝑔(π‘”β€˜π‘₯) β‰  βˆ…))
6236, 61syldc 48 . . . 4 (βˆ€π‘“(𝑓:dom π‘“βŸΆComp β†’ (∏tβ€˜π‘“) ∈ Comp) β†’ ((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) β†’ Xπ‘₯ ∈ dom 𝑔(π‘”β€˜π‘₯) β‰  βˆ…))
6362alrimiv 1930 . . 3 (βˆ€π‘“(𝑓:dom π‘“βŸΆComp β†’ (∏tβ€˜π‘“) ∈ Comp) β†’ βˆ€π‘”((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) β†’ Xπ‘₯ ∈ dom 𝑔(π‘”β€˜π‘₯) β‰  βˆ…))
64 dfac9 10133 . . 3 (CHOICE ↔ βˆ€π‘”((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) β†’ Xπ‘₯ ∈ dom 𝑔(π‘”β€˜π‘₯) β‰  βˆ…))
6563, 64sylibr 233 . 2 (βˆ€π‘“(𝑓:dom π‘“βŸΆComp β†’ (∏tβ€˜π‘“) ∈ Comp) β†’ CHOICE)
6620, 65impbii 208 1 (CHOICE ↔ βˆ€π‘“(𝑓:dom π‘“βŸΆComp β†’ (∏tβ€˜π‘“) ∈ Comp))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 396  βˆ€wal 1539   = wceq 1541   ∈ wcel 2106   β‰  wne 2940   βˆ‰ wnel 3046  Vcvv 3474   ∩ cin 3947  βˆ…c0 4322  π’« cpw 4602  {csn 4628  {cpr 4630  βˆͺ cuni 4908   ↦ cmpt 5231  dom cdm 5676  ran crn 5677  Fun wfun 6537  βŸΆwf 6539  β€˜cfv 6543  Xcixp 8893  cardccrd 9932  CHOICEwac 10112  topGenctg 17387  βˆtcpt 17388  Compccmp 23110  UFLcufl 23624
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-rpss 7715  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-omul 8473  df-er 8705  df-map 8824  df-ixp 8894  df-en 8942  df-dom 8943  df-fin 8945  df-fi 9408  df-wdom 9562  df-dju 9898  df-card 9936  df-acn 9939  df-ac 10113  df-topgen 17393  df-pt 17394  df-fbas 21141  df-fg 21142  df-top 22616  df-topon 22633  df-bases 22669  df-cld 22743  df-ntr 22744  df-cls 22745  df-nei 22822  df-cmp 23111  df-fil 23570  df-ufil 23625  df-ufl 23626  df-flim 23663  df-fcls 23665
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator