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Theorem dfac21 41856
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 3479 . . . . . . 7 𝑓 ∈ V
21dmex 7902 . . . . . 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 6905 . . . . . . . 8 (∏tβ€˜π‘“) ∈ V
65uniex 7731 . . . . . . 7 βˆͺ (∏tβ€˜π‘“) ∈ V
7 acufl 23421 . . . . . . . 8 (CHOICE β†’ UFL = V)
87adantr 482 . . . . . . 7 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ UFL = V)
96, 8eleqtrrid 2841 . . . . . 6 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ βˆͺ (∏tβ€˜π‘“) ∈ UFL)
10 simpl 484 . . . . . . . 8 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ CHOICE)
11 dfac10 10132 . . . . . . . 8 (CHOICE ↔ dom card = V)
1210, 11sylib 217 . . . . . . 7 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ dom card = V)
136, 12eleqtrrid 2841 . . . . . 6 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ βˆͺ (∏tβ€˜π‘“) ∈ dom card)
149, 13elind 4195 . . . . 5 ((CHOICE ∧ 𝑓:dom π‘“βŸΆComp) β†’ βˆͺ (∏tβ€˜π‘“) ∈ (UFL ∩ dom card))
15 eqid 2733 . . . . . 6 (∏tβ€˜π‘“) = (∏tβ€˜π‘“)
16 eqid 2733 . . . . . 6 βˆͺ (∏tβ€˜π‘“) = βˆͺ (∏tβ€˜π‘“)
1715, 16ptcmpg 23561 . . . . 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 6905 . . . . . . . . 9 (π‘”β€˜π‘¦) ∈ V
22 kelac2lem 41854 . . . . . . . . 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 7115 . . . . . . 7 ((Fun 𝑔 ∧ βˆ… βˆ‰ ran 𝑔) β†’ (𝑦 ∈ dom 𝑔 ↦ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}})):dom π‘”βŸΆComp)
2524ffdmd 6749 . . . . . 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 3479 . . . . . . . . 9 𝑔 ∈ V
2726dmex 7902 . . . . . . . 8 dom 𝑔 ∈ V
2827mptex 7225 . . . . . . 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 5904 . . . . . . . . 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 6706 . . . . . . . 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 6892 . . . . . . . . 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 2819 . . . . . . . 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 3596 . . . . . 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 6905 . . . . . . . 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 3048 . . . . . . . . . . 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 7079 . . . . . . . . . . . 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 2822 . . . . . . . . . . 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 2955 . . . . . . . . 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 6892 . . . . . . . . . . . . . 14 (𝑦 = π‘₯ β†’ (π‘”β€˜π‘¦) = (π‘”β€˜π‘₯))
5049unieqd 4923 . . . . . . . . . . . . . . . 16 (𝑦 = π‘₯ β†’ βˆͺ (π‘”β€˜π‘¦) = βˆͺ (π‘”β€˜π‘₯))
5150pweqd 4620 . . . . . . . . . . . . . . 15 (𝑦 = π‘₯ β†’ 𝒫 βˆͺ (π‘”β€˜π‘¦) = 𝒫 βˆͺ (π‘”β€˜π‘₯))
5251sneqd 4641 . . . . . . . . . . . . . 14 (𝑦 = π‘₯ β†’ {𝒫 βˆͺ (π‘”β€˜π‘¦)} = {𝒫 βˆͺ (π‘”β€˜π‘₯)})
5349, 52preq12d 4746 . . . . . . . . . . . . 13 (𝑦 = π‘₯ β†’ {(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}} = {(π‘”β€˜π‘₯), {𝒫 βˆͺ (π‘”β€˜π‘₯)}})
5453fveq2d 6896 . . . . . . . . . . . 12 (𝑦 = π‘₯ β†’ (topGenβ€˜{(π‘”β€˜π‘¦), {𝒫 βˆͺ (π‘”β€˜π‘¦)}}) = (topGenβ€˜{(π‘”β€˜π‘₯), {𝒫 βˆͺ (π‘”β€˜π‘₯)}}))
5554cbvmptv 5262 . . . . . . . . . . 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 6895 . . . . . . . . . 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 2825 . . . . . . . . 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 41855 . . . . . 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 10131 . . 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 2941   βˆ‰ wnel 3047  Vcvv 3475   ∩ cin 3948  βˆ…c0 4323  π’« cpw 4603  {csn 4629  {cpr 4631  βˆͺ cuni 4909   ↦ cmpt 5232  dom cdm 5677  ran crn 5678  Fun wfun 6538  βŸΆwf 6540  β€˜cfv 6544  Xcixp 8891  cardccrd 9930  CHOICEwac 10110  topGenctg 17383  βˆtcpt 17384  Compccmp 22890  UFLcufl 23404
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 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-iin 5001  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-se 5633  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-isom 6553  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-rpss 7713  df-om 7856  df-1st 7975  df-2nd 7976  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-1o 8466  df-oadd 8470  df-omul 8471  df-er 8703  df-map 8822  df-ixp 8892  df-en 8940  df-dom 8941  df-fin 8943  df-fi 9406  df-wdom 9560  df-dju 9896  df-card 9934  df-acn 9937  df-ac 10111  df-topgen 17389  df-pt 17390  df-fbas 20941  df-fg 20942  df-top 22396  df-topon 22413  df-bases 22449  df-cld 22523  df-ntr 22524  df-cls 22525  df-nei 22602  df-cmp 22891  df-fil 23350  df-ufil 23405  df-ufl 23406  df-flim 23443  df-fcls 23445
This theorem is referenced by: (None)
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