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Theorem dfac21 38156
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 3343 . . . . . . 7 𝑓 ∈ V
21dmex 7265 . . . . . 6 dom 𝑓 ∈ V
32a1i 11 . . . . 5 ((CHOICE𝑓:dom 𝑓⟶Comp) → dom 𝑓 ∈ V)
4 simpr 479 . . . . 5 ((CHOICE𝑓:dom 𝑓⟶Comp) → 𝑓:dom 𝑓⟶Comp)
5 fvex 6363 . . . . . . . 8 (∏t𝑓) ∈ V
65uniex 7119 . . . . . . 7 (∏t𝑓) ∈ V
7 acufl 21942 . . . . . . . 8 (CHOICE → UFL = V)
87adantr 472 . . . . . . 7 ((CHOICE𝑓:dom 𝑓⟶Comp) → UFL = V)
96, 8syl5eleqr 2846 . . . . . 6 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ UFL)
10 simpl 474 . . . . . . . 8 ((CHOICE𝑓:dom 𝑓⟶Comp) → CHOICE)
11 dfac10 9171 . . . . . . . 8 (CHOICE ↔ dom card = V)
1210, 11sylib 208 . . . . . . 7 ((CHOICE𝑓:dom 𝑓⟶Comp) → dom card = V)
136, 12syl5eleqr 2846 . . . . . 6 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ dom card)
149, 13elind 3941 . . . . 5 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ (UFL ∩ dom card))
15 eqid 2760 . . . . . 6 (∏t𝑓) = (∏t𝑓)
16 eqid 2760 . . . . . 6 (∏t𝑓) = (∏t𝑓)
1715, 16ptcmpg 22082 . . . . 5 ((dom 𝑓 ∈ V ∧ 𝑓:dom 𝑓⟶Comp ∧ (∏t𝑓) ∈ (UFL ∩ dom card)) → (∏t𝑓) ∈ Comp)
183, 4, 14, 17syl3anc 1477 . . . 4 ((CHOICE𝑓:dom 𝑓⟶Comp) → (∏t𝑓) ∈ Comp)
1918ex 449 . . 3 (CHOICE → (𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp))
2019alrimiv 2004 . 2 (CHOICE → ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp))
21 fvex 6363 . . . . . . . . . 10 (𝑔𝑦) ∈ V
22 kelac2lem 38154 . . . . . . . . . 10 ((𝑔𝑦) ∈ V → (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}) ∈ Comp)
2321, 22mp1i 13 . . . . . . . . 9 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑦 ∈ dom 𝑔) → (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}) ∈ Comp)
24 eqid 2760 . . . . . . . . 9 (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))
2523, 24fmptd 6549 . . . . . . . 8 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})):dom 𝑔⟶Comp)
26 ffdm 6223 . . . . . . . 8 ((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 . . . . . . 7 ((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 477 . . . . . 6 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))⟶Comp)
29 vex 3343 . . . . . . . . 9 𝑔 ∈ V
3029dmex 7265 . . . . . . . 8 dom 𝑔 ∈ V
3130mptex 6651 . . . . . . 7 (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) ∈ V
32 id 22 . . . . . . . . 9 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → 𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})))
33 dmeq 5479 . . . . . . . . 9 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → dom 𝑓 = dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})))
3432, 33feq12d 6194 . . . . . . . 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))
35 fveq2 6353 . . . . . . . . 9 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → (∏t𝑓) = (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))))
3635eleq1d 2824 . . . . . . . 8 (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) → ((∏t𝑓) ∈ Comp ↔ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp))
3734, 36imbi12d 333 . . . . . . 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)))
3831, 37spcv 3439 . . . . . 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))
3928, 38syl5com 31 . . . . 5 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp))
40 fvex 6363 . . . . . . . 8 (𝑔𝑥) ∈ V
4140a1i 11 . . . . . . 7 ((((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ∈ V)
42 df-nel 3036 . . . . . . . . . . 11 (∅ ∉ ran 𝑔 ↔ ¬ ∅ ∈ ran 𝑔)
4342biimpi 206 . . . . . . . . . 10 (∅ ∉ ran 𝑔 → ¬ ∅ ∈ ran 𝑔)
4443ad2antlr 765 . . . . . . . . 9 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → ¬ ∅ ∈ ran 𝑔)
45 fvelrn 6516 . . . . . . . . . . . 12 ((Fun 𝑔𝑥 ∈ dom 𝑔) → (𝑔𝑥) ∈ ran 𝑔)
4645adantlr 753 . . . . . . . . . . 11 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ∈ ran 𝑔)
47 eleq1 2827 . . . . . . . . . . 11 ((𝑔𝑥) = ∅ → ((𝑔𝑥) ∈ ran 𝑔 ↔ ∅ ∈ ran 𝑔))
4846, 47syl5ibcom 235 . . . . . . . . . 10 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → ((𝑔𝑥) = ∅ → ∅ ∈ ran 𝑔))
4948necon3bd 2946 . . . . . . . . 9 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (¬ ∅ ∈ ran 𝑔 → (𝑔𝑥) ≠ ∅))
5044, 49mpd 15 . . . . . . . 8 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ≠ ∅)
5150adantlr 753 . . . . . . 7 ((((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔𝑥) ≠ ∅)
52 fveq2 6353 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → (𝑔𝑦) = (𝑔𝑥))
5352unieqd 4598 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑥 (𝑔𝑦) = (𝑔𝑥))
5453pweqd 4307 . . . . . . . . . . . . . . 15 (𝑦 = 𝑥 → 𝒫 (𝑔𝑦) = 𝒫 (𝑔𝑥))
5554sneqd 4333 . . . . . . . . . . . . . 14 (𝑦 = 𝑥 → {𝒫 (𝑔𝑦)} = {𝒫 (𝑔𝑥)})
5652, 55preq12d 4420 . . . . . . . . . . . . 13 (𝑦 = 𝑥 → {(𝑔𝑦), {𝒫 (𝑔𝑦)}} = {(𝑔𝑥), {𝒫 (𝑔𝑥)}})
5756fveq2d 6357 . . . . . . . . . . . 12 (𝑦 = 𝑥 → (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}) = (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))
5857cbvmptv 4902 . . . . . . . . . . 11 (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}})) = (𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))
5958fveq2i 6356 . . . . . . . . . 10 (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) = (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}})))
6059eleq1i 2830 . . . . . . . . 9 ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp ↔ (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))) ∈ Comp)
6160biimpi 206 . . . . . . . 8 ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp → (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑥), {𝒫 (𝑔𝑥)}}))) ∈ Comp)
6261adantl 473 . . . . . . 7 (((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 38155 . . . . . 6 (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅)
6463ex 449 . . . . 5 ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔𝑦), {𝒫 (𝑔𝑦)}}))) ∈ Comp → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
6539, 64syldc 48 . . . 4 (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
6665alrimiv 2004 . . 3 (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
67 dfac9 9170 . . 3 (CHOICE ↔ ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔𝑥) ≠ ∅))
6866, 67sylibr 224 . 2 (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp) → CHOICE)
6920, 68impbii 199 1 (CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t𝑓) ∈ Comp))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  wal 1630   = wceq 1632  wcel 2139  wne 2932  wnel 3035  Vcvv 3340  cin 3714  wss 3715  c0 4058  𝒫 cpw 4302  {csn 4321  {cpr 4323   cuni 4588  cmpt 4881  dom cdm 5266  ran crn 5267  Fun wfun 6043  wf 6045  cfv 6049  Xcixp 8076  cardccrd 8971  CHOICEwac 9148  topGenctg 16320  tcpt 16321  Compccmp 21411  UFLcufl 21925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-rpss 7103  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-2o 7731  df-oadd 7734  df-omul 7735  df-er 7913  df-map 8027  df-ixp 8077  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-fi 8484  df-wdom 8631  df-card 8975  df-acn 8978  df-ac 9149  df-cda 9202  df-topgen 16326  df-pt 16327  df-fbas 19965  df-fg 19966  df-top 20921  df-topon 20938  df-bases 20972  df-cld 21045  df-ntr 21046  df-cls 21047  df-nei 21124  df-cmp 21412  df-fil 21871  df-ufil 21926  df-ufl 21927  df-flim 21964  df-fcls 21966
This theorem is referenced by: (None)
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