Theorem dfac21 43511

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 → (∏t‘𝑓) ∈ Comp))

Proof of Theorem dfac21

Dummy variables 𝑔 𝑦 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3435	. . . . . . 7 ⊢ 𝑓 ∈ V
2	1	dmex 7849	. . . . . 6 ⊢ dom 𝑓 ∈ V
3	2	a1i 11	. . . . 5 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → dom 𝑓 ∈ V)
4		simpr 485	. . . . 5 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → 𝑓:dom 𝑓⟶Comp)
5		fvex 6840	. . . . . . . 8 ⊢ (∏t‘𝑓) ∈ V
6	5	uniex 7684	. . . . . . 7 ⊢ ∪ (∏t‘𝑓) ∈ V
7		acufl 23900	. . . . . . . 8 ⊢ (CHOICE → UFL = V)
8	7	adantr 481	. . . . . . 7 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → UFL = V)
9	6, 8	eleqtrrid 2846	. . . . . 6 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → ∪ (∏t‘𝑓) ∈ UFL)
10		dfac10 10051	. . . . . . . 8 ⊢ (CHOICE ↔ dom card = V)
11	10	birani 504	. . . . . . 7 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → dom card = V)
12	6, 11	eleqtrrid 2846	. . . . . 6 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → ∪ (∏t‘𝑓) ∈ dom card)
13	9, 12	elind 4129	. . . . 5 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → ∪ (∏t‘𝑓) ∈ (UFL ∩ dom card))
14		eqid 2739	. . . . . 6 ⊢ (∏t‘𝑓) = (∏t‘𝑓)
15		eqid 2739	. . . . . 6 ⊢ ∪ (∏t‘𝑓) = ∪ (∏t‘𝑓)
16	14, 15	ptcmpg 24040	. . . . 5 ⊢ ((dom 𝑓 ∈ V ∧ 𝑓:dom 𝑓⟶Comp ∧ ∪ (∏t‘𝑓) ∈ (UFL ∩ dom card)) → (∏t‘𝑓) ∈ Comp)
17	3, 4, 13, 16	syl3anc 1379	. . . 4 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → (∏t‘𝑓) ∈ Comp)
18	17	ex 413	. . 3 ⊢ (CHOICE → (𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp))
19	18	alrimiv 1934	. 2 ⊢ (CHOICE → ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp))
20		fvex 6840	. . . . . . . . 9 ⊢ (𝑔‘𝑦) ∈ V
21		kelac2lem 43509	. . . . . . . . 9 ⊢ ((𝑔‘𝑦) ∈ V → (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}) ∈ Comp)
22	20, 21	mp1i 13	. . . . . . . 8 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑦 ∈ dom 𝑔) → (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}) ∈ Comp)
23	22	fmpttd 7056	. . . . . . 7 ⊢ ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})):dom 𝑔⟶Comp)
24	23	ffdmd 6685	. . . . . 6 ⊢ ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))⟶Comp)
25		vex 3435	. . . . . . . . 9 ⊢ 𝑔 ∈ V
26	25	dmex 7849	. . . . . . . 8 ⊢ dom 𝑔 ∈ V
27	26	mptex 7167	. . . . . . 7 ⊢ (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) ∈ V
28		id 22	. . . . . . . . 9 ⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) → 𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})))
29		dmeq 5845	. . . . . . . . 9 ⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) → dom 𝑓 = dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})))
30	28, 29	feq12d 6643	. . . . . . . 8 ⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) → (𝑓:dom 𝑓⟶Comp ↔ (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))⟶Comp))
31		fveq2 6827	. . . . . . . . 9 ⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) → (∏t‘𝑓) = (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))))
32	31	eleq1d 2824	. . . . . . . 8 ⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) → ((∏t‘𝑓) ∈ Comp ↔ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp))
33	30, 32	imbi12d 345	. . . . . . 7 ⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) → ((𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp) ↔ ((𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))⟶Comp → (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp)))
34	27, 33	spcv 3543	. . . . . 6 ⊢ (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp) → ((𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})):dom (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))⟶Comp → (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp))
35	24, 34	syl5com 31	. . . . 5 ⊢ ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp) → (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp))
36		fvex 6840	. . . . . . . 8 ⊢ (𝑔‘𝑥) ∈ V
37	36	a1i 11	. . . . . . 7 ⊢ ((((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ V)
38		df-nel 3039	. . . . . . . . . . 11 ⊢ (∅ ∉ ran 𝑔 ↔ ¬ ∅ ∈ ran 𝑔)
39	38	biimpi 217	. . . . . . . . . 10 ⊢ (∅ ∉ ran 𝑔 → ¬ ∅ ∈ ran 𝑔)
40	39	ad2antlr 733	. . . . . . . . 9 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → ¬ ∅ ∈ ran 𝑔)
41		fvelrn 7017	. . . . . . . . . . . 12 ⊢ ((Fun 𝑔 ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ ran 𝑔)
42	41	adantlr 721	. . . . . . . . . . 11 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ ran 𝑔)
43		eleq1 2827	. . . . . . . . . . 11 ⊢ ((𝑔‘𝑥) = ∅ → ((𝑔‘𝑥) ∈ ran 𝑔 ↔ ∅ ∈ ran 𝑔))
44	42, 43	syl5ibcom 246	. . . . . . . . . 10 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → ((𝑔‘𝑥) = ∅ → ∅ ∈ ran 𝑔))
45	44	necon3bd 2948	. . . . . . . . 9 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (¬ ∅ ∈ ran 𝑔 → (𝑔‘𝑥) ≠ ∅))
46	40, 45	mpd 15	. . . . . . . 8 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ≠ ∅)
47	46	adantlr 721	. . . . . . 7 ⊢ ((((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ≠ ∅)
48		fveq2 6827	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑔‘𝑦) = (𝑔‘𝑥))
49	48	unieqd 4851	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ∪ (𝑔‘𝑦) = ∪ (𝑔‘𝑥))
50	49	pweqd 4546	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → 𝒫 ∪ (𝑔‘𝑦) = 𝒫 ∪ (𝑔‘𝑥))
51	50	sneqd 4567	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → {𝒫 ∪ (𝑔‘𝑦)} = {𝒫 ∪ (𝑔‘𝑥)})
52	48, 51	preq12d 4673	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → {(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}} = {(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}})
53	52	fveq2d 6831	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}) = (topGen‘{(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}}))
54	53	cbvmptv 5176	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) = (𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}}))
55	54	fveq2i 6830	. . . . . . . . 9 ⊢ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) = (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}})))
56	55	eleq1i 2830	. . . . . . . 8 ⊢ ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp ↔ (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}}))) ∈ Comp)
57	56	bilani 505	. . . . . . 7 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp) → (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}}))) ∈ Comp)
58	37, 47, 57	kelac2 43510	. . . . . 6 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅)
59	58	ex 413	. . . . 5 ⊢ ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → ((∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅))
60	35, 59	syldc 48	. . . 4 ⊢ (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp) → ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅))
61	60	alrimiv 1934	. . 3 ⊢ (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp) → ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅))
62		dfac9 10050	. . 3 ⊢ (CHOICE ↔ ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅))
63	61, 62	sylibr 235	. 2 ⊢ (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp) → CHOICE)
64	19, 63	impbii 210	1 ⊢ (CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1545   = wceq 1547   ∈ wcel 2119   ≠ wne 2934   ∉ wnel 3038  Vcvv 3431   ∩ cin 3882  ∅c0 4261  𝒫 cpw 4529  {csn 4555  {cpr 4557  ∪ cuni 4838   ↦ cmpt 5153  dom cdm 5618  ran crn 5619  Fun wfun 6479  ⟶wf 6481  ‘cfv 6485  Xcixp 8835  cardccrd 9850  CHOICEwac 10028  topGenctg 17391  ∏_tcpt 17392  Compccmp 23369  UFLcufl 23883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-iin 4924  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-rpss 7666  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-oadd 8399  df-omul 8400  df-er 8633  df-map 8765  df-ixp 8836  df-en 8884  df-dom 8885  df-fin 8887  df-fi 9314  df-wdom 9470  df-dju 9816  df-card 9854  df-acn 9857  df-ac 10029  df-topgen 17397  df-pt 17398  df-fbas 21344  df-fg 21345  df-top 22877  df-topon 22894  df-bases 22929  df-cld 23002  df-ntr 23003  df-cls 23004  df-nei 23081  df-cmp 23370  df-fil 23829  df-ufil 23884  df-ufl 23885  df-flim 23922  df-fcls 23924

This theorem is referenced by: (None)