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Theorem dfac21 43078
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 3484 . . . . . . 7 𝑓 ∈ V
21dmex 7931 . . . . . 6 dom 𝑓 ∈ V
32a1i 11 . . . . 5 ((CHOICE𝑓:dom 𝑓⟶Comp) → dom 𝑓 ∈ V)
4 simpr 484 . . . . 5 ((CHOICE𝑓:dom 𝑓⟶Comp) → 𝑓:dom 𝑓⟶Comp)
5 fvex 6919 . . . . . . . 8 (∏t𝑓) ∈ V
65uniex 7761 . . . . . . 7 (∏t𝑓) ∈ V
7 acufl 23925 . . . . . . . 8 (CHOICE → UFL = V)
87adantr 480 . . . . . . 7 ((CHOICE𝑓:dom 𝑓⟶Comp) → UFL = V)
96, 8eleqtrrid 2848 . . . . . 6 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ UFL)
10 simpl 482 . . . . . . . 8 ((CHOICE𝑓:dom 𝑓⟶Comp) → CHOICE)
11 dfac10 10178 . . . . . . . 8 (CHOICE ↔ dom card = V)
1210, 11sylib 218 . . . . . . 7 ((CHOICE𝑓:dom 𝑓⟶Comp) → dom card = V)
136, 12eleqtrrid 2848 . . . . . 6 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ dom card)
149, 13elind 4200 . . . . 5 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ (UFL ∩ dom card))
15 eqid 2737 . . . . . 6 (∏t𝑓) = (∏t𝑓)
16 eqid 2737 . . . . . 6 (∏t𝑓) = (∏t𝑓)
1715, 16ptcmpg 24065 . . . . 5 ((dom 𝑓 ∈ V ∧ 𝑓:dom 𝑓⟶Comp ∧ (∏t𝑓) ∈ (UFL ∩ dom card)) → (∏t𝑓) ∈ Comp)
183, 4, 14, 17syl3anc 1373 . . . 4 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ Comp)
1918ex 412 . . 3 (CHOICE → (𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp))
2019alrimiv 1927 . 2 (CHOICE → ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp))
21 fvex 6919 . . . . . . . . 9 (𝑔𝑦) ∈ V
22 kelac2lem 43076 . . . . . . . . 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 7135 . . . . . . 7 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})):dom 𝑔⟶Comp)
2524ffdmd 6766 . . . . . 6 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))⟶Comp)
26 vex 3484 . . . . . . . . 9 𝑔 ∈ V
2726dmex 7931 . . . . . . . 8 dom 𝑔 ∈ V
2827mptex 7243 . . . . . . 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 5914 . . . . . . . . 9 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → dom 𝑓 = dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})))
3129, 30feq12d 6724 . . . . . . . 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 6906 . . . . . . . . 9 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → (∏t𝑓) = (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))))
3332eleq1d 2826 . . . . . . . 8 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → ((∏t𝑓) ∈ Comp ↔ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp))
3431, 33imbi12d 344 . . . . . . 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 3605 . . . . . 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 6919 . . . . . . . 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 3047 . . . . . . . . . . 11 (∅ ∉ ran 𝑔 ↔ ¬ ∅ ∈ ran 𝑔)
4039biimpi 216 . . . . . . . . . 10 (∅ ∉ ran 𝑔 → ¬ ∅ ∈ ran 𝑔)
4140ad2antlr 727 . . . . . . . . 9 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → ¬ ∅ ∈ ran 𝑔)
42 fvelrn 7096 . . . . . . . . . . . 12 ((Fun 𝑔𝑥 ∈ dom 𝑔) → (𝑔𝑥) ∈ ran 𝑔)
4342adantlr 715 . . . . . . . . . . 11 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ∈ ran 𝑔)
44 eleq1 2829 . . . . . . . . . . 11 ((𝑔𝑥) = ∅ → ((𝑔𝑥) ∈ ran 𝑔 ↔ ∅ ∈ ran 𝑔))
4543, 44syl5ibcom 245 . . . . . . . . . 10 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → ((𝑔𝑥) = ∅ → ∅ ∈ ran 𝑔))
4645necon3bd 2954 . . . . . . . . 9 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (¬ ∅ ∈ ran 𝑔 → (𝑔𝑥) ≠ ∅))
4741, 46mpd 15 . . . . . . . 8 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ≠ ∅)
4847adantlr 715 . . . . . . 7 ((((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ≠ ∅)
49 fveq2 6906 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑔𝑦) = (𝑔𝑥))
5049unieqd 4920 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 (𝑔𝑦) = (𝑔𝑥))
5150pweqd 4617 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → 𝒫 (𝑔𝑦) = 𝒫 (𝑔𝑥))
5251sneqd 4638 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → {𝒫 (𝑔𝑦)} = {𝒫 (𝑔𝑥)})
5349, 52preq12d 4741 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → {(𝑔𝑦), {𝒫 (𝑔𝑦)}} = {(𝑔𝑥), {𝒫 (𝑔𝑥)}})
5453fveq2d 6910 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}) = (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))
5554cbvmptv 5255 . . . . . . . . . . 11 (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) = (𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))
5655fveq2i 6909 . . . . . . . . . 10 (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) = (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}})))
5756eleq1i 2832 . . . . . . . . 9 ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp ↔ (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))) ∈ Comp)
5857biimpi 216 . . . . . . . 8 ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp → (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))) ∈ Comp)
5958adantl 481 . . . . . . 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 43077 . . . . . 6 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅)
6160ex 412 . . . . 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 1927 . . 3 (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
64 dfac9 10177 . . 3 (CHOICE ↔ ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
6563, 64sylibr 234 . 2 (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → CHOICE)
6620, 65impbii 209 1 (CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  wal 1538   = wceq 1540  wcel 2108  wne 2940  wnel 3046  Vcvv 3480  cin 3950  c0 4333  𝒫 cpw 4600  {csn 4626  {cpr 4628   cuni 4907  cmpt 5225  dom cdm 5685  ran crn 5686  Fun wfun 6555  wf 6557  cfv 6561  Xcixp 8937  cardccrd 9975  CHOICEwac 10155  topGenctg 17482  tcpt 17483  Compccmp 23394  UFLcufl 23908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-isom 6570  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rpss 7743  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-2o 8507  df-oadd 8510  df-omul 8511  df-er 8745  df-map 8868  df-ixp 8938  df-en 8986  df-dom 8987  df-fin 8989  df-fi 9451  df-wdom 9605  df-dju 9941  df-card 9979  df-acn 9982  df-ac 10156  df-topgen 17488  df-pt 17489  df-fbas 21361  df-fg 21362  df-top 22900  df-topon 22917  df-bases 22953  df-cld 23027  df-ntr 23028  df-cls 23029  df-nei 23106  df-cmp 23395  df-fil 23854  df-ufil 23909  df-ufl 23910  df-flim 23947  df-fcls 23949
This theorem is referenced by: (None)
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