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 41422
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 3452 . . . . . . 7 𝑓 ∈ V
21dmex 7853 . . . . . 6 dom 𝑓 ∈ V
32a1i 11 . . . . 5 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ dom 𝑓 ∈ V)
4 simpr 486 . . . . 5 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ 𝑓:dom π‘“βŸΆComp)
5 fvex 6860 . . . . . . . 8 (∏tβ€˜π‘“) ∈ V
65uniex 7683 . . . . . . 7 βˆͺ (∏tβ€˜π‘“) ∈ V
7 acufl 23284 . . . . . . . 8 (CHOICE β†’ UFL = V)
87adantr 482 . . . . . . 7 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ UFL = V)
96, 8eleqtrrid 2845 . . . . . 6 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ βˆͺ (∏tβ€˜π‘“) ∈ UFL)
10 simpl 484 . . . . . . . 8 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ CHOICE)
11 dfac10 10080 . . . . . . . 8 (CHOICE ↔ dom card = V)
1210, 11sylib 217 . . . . . . 7 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ dom card = V)
136, 12eleqtrrid 2845 . . . . . 6 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ βˆͺ (∏tβ€˜π‘“) ∈ dom card)
149, 13elind 4159 . . . . 5 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ βˆͺ (∏tβ€˜π‘“) ∈ (UFL ∩ dom card))
15 eqid 2737 . . . . . 6 (∏tβ€˜π‘“) = (∏tβ€˜π‘“)
16 eqid 2737 . . . . . 6 βˆͺ (∏tβ€˜π‘“) = βˆͺ (∏tβ€˜π‘“)
1715, 16ptcmpg 23424 . . . . 5 ((dom 𝑓 ∈ V ∧ 𝑓:dom π‘“βŸΆComp ∧ βˆͺ (∏tβ€˜π‘“) ∈ (UFL ∩ dom card)) β†’ (∏tβ€˜π‘“) ∈ Comp)
183, 4, 14, 17syl3anc 1372 . . . 4 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ (∏tβ€˜π‘“) ∈ Comp)
1918ex 414 . . 3 (CHOICE β†’ (𝑓:dom π‘“βŸΆComp β†’ (∏tβ€˜π‘“) ∈ Comp))
2019alrimiv 1931 . 2 (CHOICE β†’ βˆ€π‘“(𝑓:dom π‘“βŸΆComp β†’ (∏tβ€˜π‘“) ∈ Comp))
21 fvex 6860 . . . . . . . . 9 (π‘”β€˜π‘¦) ∈ V
22 kelac2lem 41420 . . . . . . . . 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 7068 . . . . . . 7 ((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) β†’ (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})):dom π‘”βŸΆComp)
2524ffdmd 6704 . . . . . 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 3452 . . . . . . . . 9 𝑔 ∈ V
2726dmex 7853 . . . . . . . 8 dom 𝑔 ∈ V
2827mptex 7178 . . . . . . 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 5864 . . . . . . . . 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 6661 . . . . . . . 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 6847 . . . . . . . . 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 2823 . . . . . . . 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 345 . . . . . . 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 3567 . . . . . 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 6860 . . . . . . . 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 3051 . . . . . . . . . . 11 (βˆ… βˆ‰ ran 𝑔 ↔ Β¬ βˆ… ∈ ran 𝑔)
4039biimpi 215 . . . . . . . . . 10 (βˆ… βˆ‰ ran 𝑔 β†’ Β¬ βˆ… ∈ ran 𝑔)
4140ad2antlr 726 . . . . . . . . 9 (((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) ∧ π‘₯ ∈ dom 𝑔) β†’ Β¬ βˆ… ∈ ran 𝑔)
42 fvelrn 7032 . . . . . . . . . . . 12 ((Fun 𝑔 ∧ π‘₯ ∈ dom 𝑔) β†’ (π‘”β€˜π‘₯) ∈ ran 𝑔)
4342adantlr 714 . . . . . . . . . . 11 (((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) ∧ π‘₯ ∈ dom 𝑔) β†’ (π‘”β€˜π‘₯) ∈ ran 𝑔)
44 eleq1 2826 . . . . . . . . . . 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 2958 . . . . . . . . 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 714 . . . . . . 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 6847 . . . . . . . . . . . . . 14 (𝑦 = π‘₯ β†’ (π‘”β€˜π‘¦) = (π‘”β€˜π‘₯))
5049unieqd 4884 . . . . . . . . . . . . . . . 16 (𝑦 = π‘₯ β†’ βˆͺ (π‘”β€˜π‘¦) = βˆͺ (π‘”β€˜π‘₯))
5150pweqd 4582 . . . . . . . . . . . . . . 15 (𝑦 = π‘₯ β†’ 𝒫 βˆͺ (π‘”β€˜π‘¦) = 𝒫 βˆͺ (π‘”β€˜π‘₯))
5251sneqd 4603 . . . . . . . . . . . . . 14 (𝑦 = π‘₯ β†’ {𝒫 βˆͺ (π‘”β€˜π‘¦)} = {𝒫 βˆͺ (π‘”β€˜π‘₯)})
5349, 52preq12d 4707 . . . . . . . . . . . . 13 (𝑦 = π‘₯ β†’ {(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}} = {(π‘”β€˜π‘₯), {𝒫 βˆͺ (π‘”β€˜π‘₯)}})
5453fveq2d 6851 . . . . . . . . . . . 12 (𝑦 = π‘₯ β†’ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}) = (topGenβ€˜{(π‘”β€˜π‘₯), {𝒫 βˆͺ (π‘”β€˜π‘₯)}}))
5554cbvmptv 5223 . . . . . . . . . . 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 6850 . . . . . . . . . 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 2829 . . . . . . . . 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 483 . . . . . . 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 41421 . . . . . 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 414 . . . . 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 1931 . . 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 10079 . . 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 397  βˆ€wal 1540   = wceq 1542   ∈ wcel 2107   β‰  wne 2944   βˆ‰ wnel 3050  Vcvv 3448   ∩ cin 3914  βˆ…c0 4287  π’« cpw 4565  {csn 4591  {cpr 4593  βˆͺ cuni 4870   ↦ cmpt 5193  dom cdm 5638  ran crn 5639  Fun wfun 6495  βŸΆwf 6497  β€˜cfv 6501  Xcixp 8842  cardccrd 9878  CHOICEwac 10058  topGenctg 17326  βˆtcpt 17327  Compccmp 22753  UFLcufl 23267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-nel 3051  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-iin 4962  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-rpss 7665  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-oadd 8421  df-omul 8422  df-er 8655  df-map 8774  df-ixp 8843  df-en 8891  df-dom 8892  df-fin 8894  df-fi 9354  df-wdom 9508  df-dju 9844  df-card 9882  df-acn 9885  df-ac 10059  df-topgen 17332  df-pt 17333  df-fbas 20809  df-fg 20810  df-top 22259  df-topon 22276  df-bases 22312  df-cld 22386  df-ntr 22387  df-cls 22388  df-nei 22465  df-cmp 22754  df-fil 23213  df-ufil 23268  df-ufl 23269  df-flim 23306  df-fcls 23308
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator