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 39659
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 ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp))

Proof of Theorem dfac21
Dummy variables 𝑔 𝑦 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3498 . . . . . . 7 𝑓 ∈ V
21dmex 7610 . . . . . 6 dom 𝑓 ∈ V
32a1i 11 . . . . 5 ((CHOICE𝑓:dom 𝑓⟶Comp) → dom 𝑓 ∈ V)
4 simpr 487 . . . . 5 ((CHOICE𝑓:dom 𝑓⟶Comp) → 𝑓:dom 𝑓⟶Comp)
5 fvex 6678 . . . . . . . 8 (∏t𝑓) ∈ V
65uniex 7461 . . . . . . 7 (∏t𝑓) ∈ V
7 acufl 22519 . . . . . . . 8 (CHOICE → UFL = V)
87adantr 483 . . . . . . 7 ((CHOICE𝑓:dom 𝑓⟶Comp) → UFL = V)
96, 8eleqtrrid 2920 . . . . . 6 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ UFL)
10 simpl 485 . . . . . . . 8 ((CHOICE𝑓:dom 𝑓⟶Comp) → CHOICE)
11 dfac10 9557 . . . . . . . 8 (CHOICE ↔ dom card = V)
1210, 11sylib 220 . . . . . . 7 ((CHOICE𝑓:dom 𝑓⟶Comp) → dom card = V)
136, 12eleqtrrid 2920 . . . . . 6 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ dom card)
149, 13elind 4171 . . . . 5 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ (UFL ∩ dom card))
15 eqid 2821 . . . . . 6 (∏t𝑓) = (∏t𝑓)
16 eqid 2821 . . . . . 6 (∏t𝑓) = (∏t𝑓)
1715, 16ptcmpg 22659 . . . . 5 ((dom 𝑓 ∈ V ∧ 𝑓:dom 𝑓⟶Comp ∧ (∏t𝑓) ∈ (UFL ∩ dom card)) → (∏t𝑓) ∈ Comp)
183, 4, 14, 17syl3anc 1367 . . . 4 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ Comp)
1918ex 415 . . 3 (CHOICE → (𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp))
2019alrimiv 1924 . 2 (CHOICE → ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp))
21 fvex 6678 . . . . . . . . 9 (𝑔𝑦) ∈ V
22 kelac2lem 39657 . . . . . . . . 9 ((𝑔𝑦) ∈ V → (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}) ∈ Comp)
2321, 22mp1i 13 . . . . . . . 8 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑦 ∈ dom 𝑔) → (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}) ∈ Comp)
2423fmpttd 6874 . . . . . . 7 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})):dom 𝑔⟶Comp)
2524ffdmd 6532 . . . . . 6 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))⟶Comp)
26 vex 3498 . . . . . . . . 9 𝑔 ∈ V
2726dmex 7610 . . . . . . . 8 dom 𝑔 ∈ V
2827mptex 6980 . . . . . . 7 (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) ∈ V
29 id 22 . . . . . . . . 9 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → 𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})))
30 dmeq 5767 . . . . . . . . 9 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → dom 𝑓 = dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})))
3129, 30feq12d 6497 . . . . . . . 8 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → (𝑓:dom 𝑓⟶Comp ↔ (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))⟶Comp))
32 fveq2 6665 . . . . . . . . 9 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → (∏t𝑓) = (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))))
3332eleq1d 2897 . . . . . . . 8 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → ((∏t𝑓) ∈ Comp ↔ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp))
3431, 33imbi12d 347 . . . . . . 7 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → ((𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) ↔ ((𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))⟶Comp → (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp)))
3528, 34spcv 3606 . . . . . 6 (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → ((𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))⟶Comp → (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp))
3625, 35syl5com 31 . . . . 5 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp))
37 fvex 6678 . . . . . . . 8 (𝑔𝑥) ∈ V
3837a1i 11 . . . . . . 7 ((((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ∈ V)
39 df-nel 3124 . . . . . . . . . . 11 (∅ ∉ ran 𝑔 ↔ ¬ ∅ ∈ ran 𝑔)
4039biimpi 218 . . . . . . . . . 10 (∅ ∉ ran 𝑔 → ¬ ∅ ∈ ran 𝑔)
4140ad2antlr 725 . . . . . . . . 9 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → ¬ ∅ ∈ ran 𝑔)
42 fvelrn 6839 . . . . . . . . . . . 12 ((Fun 𝑔𝑥 ∈ dom 𝑔) → (𝑔𝑥) ∈ ran 𝑔)
4342adantlr 713 . . . . . . . . . . 11 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ∈ ran 𝑔)
44 eleq1 2900 . . . . . . . . . . 11 ((𝑔𝑥) = ∅ → ((𝑔𝑥) ∈ ran 𝑔 ↔ ∅ ∈ ran 𝑔))
4543, 44syl5ibcom 247 . . . . . . . . . 10 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → ((𝑔𝑥) = ∅ → ∅ ∈ ran 𝑔))
4645necon3bd 3030 . . . . . . . . 9 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (¬ ∅ ∈ ran 𝑔 → (𝑔𝑥) ≠ ∅))
4741, 46mpd 15 . . . . . . . 8 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ≠ ∅)
4847adantlr 713 . . . . . . 7 ((((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ≠ ∅)
49 fveq2 6665 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑔𝑦) = (𝑔𝑥))
5049unieqd 4842 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 (𝑔𝑦) = (𝑔𝑥))
5150pweqd 4544 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → 𝒫 (𝑔𝑦) = 𝒫 (𝑔𝑥))
5251sneqd 4573 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → {𝒫 (𝑔𝑦)} = {𝒫 (𝑔𝑥)})
5349, 52preq12d 4671 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → {(𝑔𝑦), {𝒫 (𝑔𝑦)}} = {(𝑔𝑥), {𝒫 (𝑔𝑥)}})
5453fveq2d 6669 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}) = (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))
5554cbvmptv 5162 . . . . . . . . . . 11 (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) = (𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))
5655fveq2i 6668 . . . . . . . . . 10 (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) = (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}})))
5756eleq1i 2903 . . . . . . . . 9 ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp ↔ (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))) ∈ Comp)
5857biimpi 218 . . . . . . . 8 ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp → (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))) ∈ Comp)
5958adantl 484 . . . . . . 7 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp) → (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))) ∈ Comp)
6038, 48, 59kelac2 39658 . . . . . 6 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅)
6160ex 415 . . . . 5 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
6236, 61syldc 48 . . . 4 (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
6362alrimiv 1924 . . 3 (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
64 dfac9 9556 . . 3 (CHOICE ↔ ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
6563, 64sylibr 236 . 2 (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → CHOICE)
6620, 65impbii 211 1 (CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1531   = wceq 1533  wcel 2110  wne 3016  wnel 3123  Vcvv 3495  cin 3935  c0 4291  𝒫 cpw 4539  {csn 4561  {cpr 4563   cuni 4832  cmpt 5139  dom cdm 5550  ran crn 5551  Fun wfun 6344  wf 6346  cfv 6350  Xcixp 8455  cardccrd 9358  CHOICEwac 9535  topGenctg 16705  tcpt 16706  Compccmp 21988  UFLcufl 22502
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-int 4870  df-iun 4914  df-iin 4915  df-br 5060  df-opab 5122  df-mpt 5140  df-tr 5166  df-id 5455  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-se 5510  df-we 5511  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-pred 6143  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-isom 6359  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-rpss 7443  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-1o 8096  df-2o 8097  df-oadd 8100  df-omul 8101  df-er 8283  df-map 8402  df-ixp 8456  df-en 8504  df-dom 8505  df-sdom 8506  df-fin 8507  df-fi 8869  df-wdom 9017  df-dju 9324  df-card 9362  df-acn 9365  df-ac 9536  df-topgen 16711  df-pt 16712  df-fbas 20536  df-fg 20537  df-top 21496  df-topon 21513  df-bases 21548  df-cld 21621  df-ntr 21622  df-cls 21623  df-nei 21700  df-cmp 21989  df-fil 22448  df-ufil 22503  df-ufl 22504  df-flim 22541  df-fcls 22543
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator