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Theorem dfac21 36654
 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 3176 . . . . . . 7 𝑓 ∈ V
21dmex 6991 . . . . . 6 dom 𝑓 ∈ V
32a1i 11 . . . . 5 ((CHOICE𝑓:dom 𝑓⟶Comp) → dom 𝑓 ∈ V)
4 simpr 476 . . . . 5 ((CHOICE𝑓:dom 𝑓⟶Comp) → 𝑓:dom 𝑓⟶Comp)
5 fvex 6113 . . . . . . . 8 (∏t𝑓) ∈ V
65uniex 6851 . . . . . . 7 (∏t𝑓) ∈ V
7 acufl 21531 . . . . . . . 8 (CHOICE → UFL = V)
87adantr 480 . . . . . . 7 ((CHOICE𝑓:dom 𝑓⟶Comp) → UFL = V)
96, 8syl5eleqr 2695 . . . . . 6 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ UFL)
10 simpl 472 . . . . . . . 8 ((CHOICE𝑓:dom 𝑓⟶Comp) → CHOICE)
11 dfac10 8842 . . . . . . . 8 (CHOICE ↔ dom card = V)
1210, 11sylib 207 . . . . . . 7 ((CHOICE𝑓:dom 𝑓⟶Comp) → dom card = V)
136, 12syl5eleqr 2695 . . . . . 6 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ dom card)
149, 13elind 3760 . . . . 5 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ (UFL ∩ dom card))
15 eqid 2610 . . . . . 6 (∏t𝑓) = (∏t𝑓)
16 eqid 2610 . . . . . 6 (∏t𝑓) = (∏t𝑓)
1715, 16ptcmpg 21671 . . . . 5 ((dom 𝑓 ∈ V ∧ 𝑓:dom 𝑓⟶Comp ∧ (∏t𝑓) ∈ (UFL ∩ dom card)) → (∏t𝑓) ∈ Comp)
183, 4, 14, 17syl3anc 1318 . . . 4 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ Comp)
1918ex 449 . . 3 (CHOICE → (𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp))
2019alrimiv 1842 . 2 (CHOICE → ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp))
21 fvex 6113 . . . . . . . . . . 11 (𝑔𝑦) ∈ V
22 kelac2lem 36652 . . . . . . . . . . 11 ((𝑔𝑦) ∈ V → (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}) ∈ Comp)
2321, 22mp1i 13 . . . . . . . . . 10 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑦 ∈ dom 𝑔) → (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}) ∈ Comp)
24 eqid 2610 . . . . . . . . . 10 (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))
2523, 24fmptd 6292 . . . . . . . . 9 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})):dom 𝑔⟶Comp)
26 ffdm 5975 . . . . . . . . 9 ((𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})):dom 𝑔⟶Comp → ((𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))⟶Comp ∧ dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) ⊆ dom 𝑔))
2725, 26syl 17 . . . . . . . 8 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → ((𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))⟶Comp ∧ dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) ⊆ dom 𝑔))
2827simpld 474 . . . . . . 7 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))⟶Comp)
29 vex 3176 . . . . . . . . . 10 𝑔 ∈ V
3029dmex 6991 . . . . . . . . 9 dom 𝑔 ∈ V
3130mptex 6390 . . . . . . . 8 (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) ∈ V
32 id 22 . . . . . . . . . 10 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → 𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})))
33 dmeq 5246 . . . . . . . . . 10 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → dom 𝑓 = dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})))
3432, 33feq12d 5946 . . . . . . . . 9 (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))
35 fveq2 6103 . . . . . . . . . 10 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → (∏t𝑓) = (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))))
3635eleq1d 2672 . . . . . . . . 9 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → ((∏t𝑓) ∈ Comp ↔ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp))
3734, 36imbi12d 333 . . . . . . . 8 (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)))
3831, 37spcv 3272 . . . . . . 7 (∀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))
3928, 38syl5com 31 . . . . . 6 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp))
40 fvex 6113 . . . . . . . . 9 (𝑔𝑥) ∈ V
4140a1i 11 . . . . . . . 8 ((((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ∈ V)
42 df-nel 2783 . . . . . . . . . . . 12 (∅ ∉ ran 𝑔 ↔ ¬ ∅ ∈ ran 𝑔)
4342biimpi 205 . . . . . . . . . . 11 (∅ ∉ ran 𝑔 → ¬ ∅ ∈ ran 𝑔)
4443ad2antlr 759 . . . . . . . . . 10 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → ¬ ∅ ∈ ran 𝑔)
45 fvelrn 6260 . . . . . . . . . . . . 13 ((Fun 𝑔𝑥 ∈ dom 𝑔) → (𝑔𝑥) ∈ ran 𝑔)
4645adantlr 747 . . . . . . . . . . . 12 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ∈ ran 𝑔)
47 eleq1 2676 . . . . . . . . . . . 12 ((𝑔𝑥) = ∅ → ((𝑔𝑥) ∈ ran 𝑔 ↔ ∅ ∈ ran 𝑔))
4846, 47syl5ibcom 234 . . . . . . . . . . 11 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → ((𝑔𝑥) = ∅ → ∅ ∈ ran 𝑔))
4948necon3bd 2796 . . . . . . . . . 10 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (¬ ∅ ∈ ran 𝑔 → (𝑔𝑥) ≠ ∅))
5044, 49mpd 15 . . . . . . . . 9 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ≠ ∅)
5150adantlr 747 . . . . . . . 8 ((((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ≠ ∅)
52 fveq2 6103 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → (𝑔𝑦) = (𝑔𝑥))
5352unieqd 4382 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑥 (𝑔𝑦) = (𝑔𝑥))
5453pweqd 4113 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 → 𝒫 (𝑔𝑦) = 𝒫 (𝑔𝑥))
5554sneqd 4137 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → {𝒫 (𝑔𝑦)} = {𝒫 (𝑔𝑥)})
5652, 55preq12d 4220 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → {(𝑔𝑦), {𝒫 (𝑔𝑦)}} = {(𝑔𝑥), {𝒫 (𝑔𝑥)}})
5756fveq2d 6107 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}) = (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))
5857cbvmptv 4678 . . . . . . . . . . . 12 (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) = (𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))
5958fveq2i 6106 . . . . . . . . . . 11 (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) = (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}})))
6059eleq1i 2679 . . . . . . . . . 10 ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp ↔ (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))) ∈ Comp)
6160biimpi 205 . . . . . . . . 9 ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp → (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))) ∈ Comp)
6261adantl 481 . . . . . . . 8 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp) → (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))) ∈ Comp)
6341, 51, 62kelac2 36653 . . . . . . 7 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅)
6463ex 449 . . . . . 6 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
6539, 64syld 46 . . . . 5 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
6665com12 32 . . . 4 (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
6766alrimiv 1842 . . 3 (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
68 dfac9 8841 . . 3 (CHOICE ↔ ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
6967, 68sylibr 223 . 2 (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → CHOICE)
7020, 69impbii 198 1 (CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475   ∈ wcel 1977   ≠ wne 2780   ∉ wnel 2781  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  {cpr 4127  ∪ cuni 4372   ↦ cmpt 4643  dom cdm 5038  ran crn 5039  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  Xcixp 7794  cardccrd 8644  CHOICEwac 8821  topGenctg 15921  ∏tcpt 15922  Compccmp 20999  UFLcufl 21514 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-iin 4458  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-rpss 6835  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-omul 7452  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-fi 8200  df-wdom 8347  df-card 8648  df-acn 8651  df-ac 8822  df-cda 8873  df-topgen 15927  df-pt 15928  df-fbas 19564  df-fg 19565  df-top 20521  df-bases 20522  df-topon 20523  df-cld 20633  df-ntr 20634  df-cls 20635  df-nei 20712  df-cmp 21000  df-fil 21460  df-ufil 21515  df-ufl 21516  df-flim 21553  df-fcls 21555 This theorem is referenced by: (None)
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