Theorem dfac21 44067

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 3455	. . . . . . 7 ⊢ 𝑓 ∈ V
2	1	dmex 7921	. . . . . 6 ⊢ dom 𝑓 ∈ V
3	2	a1i 11	. . . . 5 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → dom 𝑓 ∈ V)
4		simpr 490	. . . . 5 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → 𝑓:dom 𝑓⟶Comp)
5		fvex 6898	. . . . . . . 8 ⊢ (∏t‘𝑓) ∈ V
6	5	uniex 7758	. . . . . . 7 ⊢ ∪ (∏t‘𝑓) ∈ V
7		acufl 24236	. . . . . . . 8 ⊢ (CHOICE → UFL = V)
8	7	adantr 486	. . . . . . 7 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → UFL = V)
9	6, 8	eleqtrrid 2868	. . . . . 6 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → ∪ (∏t‘𝑓) ∈ UFL)
10		dfac10 10216	. . . . . . . 8 ⊢ (CHOICE ↔ dom card = V)
11	10	birani 509	. . . . . . 7 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → dom card = V)
12	6, 11	eleqtrrid 2868	. . . . . 6 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → ∪ (∏t‘𝑓) ∈ dom card)
13	9, 12	elind 4146	. . . . 5 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → ∪ (∏t‘𝑓) ∈ (UFL ∩ dom card))
14		eqid 2761	. . . . . 6 ⊢ (∏t‘𝑓) = (∏t‘𝑓)
15		eqid 2761	. . . . . 6 ⊢ ∪ (∏t‘𝑓) = ∪ (∏t‘𝑓)
16	14, 15	ptcmpg 24376	. . . . 5 ⊢ ((dom 𝑓 ∈ V ∧ 𝑓:dom 𝑓⟶Comp ∧ ∪ (∏t‘𝑓) ∈ (UFL ∩ dom card)) → (∏t‘𝑓) ∈ Comp)
17	3, 4, 13, 16	syl3anc 1398	. . . 4 ⊢ ((CHOICE ∧ 𝑓:dom 𝑓⟶Comp) → (∏t‘𝑓) ∈ Comp)
18	17	ex 418	. . 3 ⊢ (CHOICE → (𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp))
19	18	alrimiv 1960	. 2 ⊢ (CHOICE → ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp))
20		fvex 6898	. . . . . . . . 9 ⊢ (𝑔‘𝑦) ∈ V
21		kelac2lem 44065	. . . . . . . . 9 ⊢ ((𝑔‘𝑦) ∈ V → (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}) ∈ Comp)
22	20, 21	mp1i 14	. . . . . . . 8 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑦 ∈ dom 𝑔) → (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}) ∈ Comp)
23	22	fmpttd 7115	. . . . . . 7 ⊢ ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})):dom 𝑔⟶Comp)
24	23	ffdmd 6740	. . . . . 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 3455	. . . . . . . . 9 ⊢ 𝑔 ∈ V
26	25	dmex 7921	. . . . . . . 8 ⊢ dom 𝑔 ∈ V
27	26	mptex 7229	. . . . . . 7 ⊢ (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) ∈ V
28		id 23	. . . . . . . . 9 ⊢ (𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) → 𝑓 = (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})))
29		dmeq 5885	. . . . . . . . 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 6697	. . . . . . . 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 6885	. . . . . . . . 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 2846	. . . . . . . 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 347	. . . . . . 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 3560	. . . . . 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 32	. . . . 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 6898	. . . . . . . 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 3063	. . . . . . . . . . 11 ⊢ (∅ ∉ ran 𝑔 ↔ ¬ ∅ ∈ ran 𝑔)
39	38	biimpi 219	. . . . . . . . . 10 ⊢ (∅ ∉ ran 𝑔 → ¬ ∅ ∈ ran 𝑔)
40	39	ad2antlr 740	. . . . . . . . 9 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → ¬ ∅ ∈ ran 𝑔)
41		fvelrn 7076	. . . . . . . . . . . 12 ⊢ ((Fun 𝑔 ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ ran 𝑔)
42	41	adantlr 728	. . . . . . . . . . 11 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ∈ ran 𝑔)
43		eleq1 2849	. . . . . . . . . . 11 ⊢ ((𝑔‘𝑥) = ∅ → ((𝑔‘𝑥) ∈ ran 𝑔 ↔ ∅ ∈ ran 𝑔))
44	42, 43	syl5ibcom 248	. . . . . . . . . 10 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → ((𝑔‘𝑥) = ∅ → ∅ ∈ ran 𝑔))
45	44	necon3bd 2970	. . . . . . . . 9 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (¬ ∅ ∈ ran 𝑔 → (𝑔‘𝑥) ≠ ∅))
46	40, 45	mpd 16	. . . . . . . 8 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ≠ ∅)
47	46	adantlr 728	. . . . . . 7 ⊢ ((((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp) ∧ 𝑥 ∈ dom 𝑔) → (𝑔‘𝑥) ≠ ∅)
48		fveq2 6885	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → (𝑔‘𝑦) = (𝑔‘𝑥))
49	48	unieqd 4880	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑥 → ∪ (𝑔‘𝑦) = ∪ (𝑔‘𝑥))
50	49	pweqd 4574	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → 𝒫 ∪ (𝑔‘𝑦) = 𝒫 ∪ (𝑔‘𝑥))
51	50	sneqd 4596	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → {𝒫 ∪ (𝑔‘𝑦)} = {𝒫 ∪ (𝑔‘𝑥)})
52	48, 51	preq12d 4702	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → {(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}} = {(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}})
53	52	fveq2d 6889	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}) = (topGen‘{(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}}))
54	53	cbvmptv 5209	. . . . . . . . . 10 ⊢ (𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}})) = (𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}}))
55	54	fveq2i 6888	. . . . . . . . 9 ⊢ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) = (∏t‘(𝑥 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑥), {𝒫 ∪ (𝑔‘𝑥)}})))
56	55	eleq1i 2852	. . . . . . . 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 510	. . . . . . 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 44066	. . . . . 6 ⊢ (((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) ∧ (∏t‘(𝑦 ∈ dom 𝑔 ↦ (topGen‘{(𝑔‘𝑦), {𝒫 ∪ (𝑔‘𝑦)}}))) ∈ Comp) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅)
59	58	ex 418	. . . . 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 49	. . . 4 ⊢ (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp) → ((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅))
61	60	alrimiv 1960	. . 3 ⊢ (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp) → ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅))
62		dfac9 10215	. . 3 ⊢ (CHOICE ↔ ∀𝑔((Fun 𝑔 ∧ ∅ ∉ ran 𝑔) → X𝑥 ∈ dom 𝑔(𝑔‘𝑥) ≠ ∅))
63	61, 62	sylibr 237	. 2 ⊢ (∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp) → CHOICE)
64	19, 63	impbii 212	1 ⊢ (CHOICE ↔ ∀𝑓(𝑓:dom 𝑓⟶Comp → (∏t‘𝑓) ∈ Comp))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145   ≠ wne 2956   ∉ wnel 3062  Vcvv 3451   ∩ cin 3898  ∅c0 4279  𝒫 cpw 4557  {csn 4584  {cpr 4586  ∪ cuni 4867   ↦ cmpt 5186  dom cdm 5651  ran crn 5652  Fun wfun 6532  ⟶wf 6534  ‘cfv 6538  Xcixp 8925  cardccrd 10016  CHOICEwac 10194  topGenctg 17608  ∏_tcpt 17609  Compccmp 23704  UFLcufl 24219

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-se 5605  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7377  df-ov 7423  df-oprab 7424  df-mpo 7425  df-rpss 7739  df-om 7878  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-1o 8476  df-2o 8477  df-oadd 8480  df-omul 8481  df-er 8717  df-map 8849  df-ixp 8926  df-en 8974  df-dom 8975  df-fin 8977  df-fi 9403  df-wdom 9559  df-dju 9982  df-card 10020  df-acn 10023  df-ac 10195  df-topgen 17614  df-pt 17615  df-fbas 21675  df-fg 21676  df-top 23212  df-topon 23229  df-bases 23264  df-cld 23337  df-ntr 23338  df-cls 23339  df-nei 23416  df-cmp 23705  df-fil 24165  df-ufil 24220  df-ufl 24221  df-flim 24258  df-fcls 24260

This theorem is used by: (None)