Theorem ptcmpg 24090

Description: Tychonoff's theorem: The product of compact spaces is compact. The choice principles needed are encoded in the last hypothesis: the base set of the product must be well-orderable and satisfy the ultrafilter lemma. Both these assumptions are satisfied if 𝒫 𝒫 𝑋 is well-orderable, so if we assume the Axiom of Choice we can eliminate them (see ptcmp 24091). (Contributed by Mario Carneiro, 27-Aug-2015.)

Hypotheses

Ref	Expression
ptcmpg.1	⊢ 𝐽 = (∏t‘𝐹)
ptcmpg.2	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
ptcmpg	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐽 ∈ Comp)

Proof of Theorem ptcmpg

Dummy variables 𝑎 𝑏 𝑘 𝑚 𝑛 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ptcmpg.1	. 2 ⊢ 𝐽 = (∏t‘𝐹)
2		nfcv 2905	. . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑎)
3		nfcv 2905	. . . 4 ⊢ Ⅎ𝑎(𝐹‘𝑘)
4		nfcv 2905	. . . 4 ⊢ Ⅎ𝑘(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏)
5		nfcv 2905	. . . 4 ⊢ Ⅎ𝑢(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏)
6		nfcv 2905	. . . 4 ⊢ Ⅎ𝑎(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)
7		nfcv 2905	. . . 4 ⊢ Ⅎ𝑏(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)
8		fveq2 6914	. . . 4 ⊢ (𝑎 = 𝑘 → (𝐹‘𝑎) = (𝐹‘𝑘))
9		fveq2 6914	. . . . . . . 8 ⊢ (𝑎 = 𝑘 → (𝑤‘𝑎) = (𝑤‘𝑘))
10	9	mpteq2dv 5253	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) = (𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)))
11	10	cnveqd 5893	. . . . . 6 ⊢ (𝑎 = 𝑘 → ◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) = ◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)))
12	11	imaeq1d 6084	. . . . 5 ⊢ (𝑎 = 𝑘 → (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏) = (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑏))
13		imaeq2 6081	. . . . 5 ⊢ (𝑏 = 𝑢 → (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑏) = (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢))
14	12, 13	sylan9eq 2797	. . . 4 ⊢ ((𝑎 = 𝑘 ∧ 𝑏 = 𝑢) → (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏) = (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢))
15	2, 3, 4, 5, 6, 7, 8, 14	cbvmpox 7533	. . 3 ⊢ (𝑎 ∈ 𝐴, 𝑏 ∈ (𝐹‘𝑎) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏)) = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢))
16		fveq2 6914	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
17	16	unieqd 4928	. . . 4 ⊢ (𝑛 = 𝑚 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑚))
18	17	cbvixpv 8963	. . 3 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = X𝑚 ∈ 𝐴 ∪ (𝐹‘𝑚)
19		simp1 1137	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐴 ∈ 𝑉)
20		simp2 1138	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐹:𝐴⟶Comp)
21		cmptop 23428	. . . . . . . 8 ⊢ (𝑘 ∈ Comp → 𝑘 ∈ Top)
22	21	ssriv 4002	. . . . . . 7 ⊢ Comp ⊆ Top
23		fss 6760	. . . . . . 7 ⊢ ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
24	20, 22, 23	sylancl 586	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐹:𝐴⟶Top)
25	1	ptuni 23627	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ 𝐽)
26	19, 24, 25	syl2anc 584	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ 𝐽)
27		ptcmpg.2	. . . . 5 ⊢ 𝑋 = ∪ 𝐽
28	26, 27	eqtr4di 2795	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = 𝑋)
29		simp3 1139	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝑋 ∈ (UFL ∩ dom card))
30	28, 29	eqeltrd 2841	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ (UFL ∩ dom card))
31	15, 18, 19, 20, 30	ptcmplem5 24089	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → (∏t‘𝐹) ∈ Comp)
32	1, 31	eqeltrid 2845	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐽 ∈ Comp)

Syntax hints:   → wi 4   ∧ w3a 1087   = wceq 1539   ∈ wcel 2108   ∩ cin 3965   ⊆ wss 3966  ∪ cuni 4915   ↦ cmpt 5234  ^◡ccnv 5692  dom cdm 5693   “ cima 5696  ⟶wf 6565  ‘cfv 6569   ∈ cmpo 7440  Xcixp 8945  cardccrd 9982  ∏_tcpt 17494  Topctop 22924  Compccmp 23419  UFLcufl 23933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  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 5288  ax-sep 5305  ax-nul 5315  ax-pow 5374  ax-pr 5441  ax-un 7761

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  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 3483  df-sbc 3795  df-csb 3912  df-dif 3969  df-un 3971  df-in 3973  df-ss 3983  df-pss 3986  df-nul 4343  df-if 4535  df-pw 4610  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-int 4955  df-iun 5001  df-iin 5002  df-br 5152  df-opab 5214  df-mpt 5235  df-tr 5269  df-id 5587  df-eprel 5593  df-po 5601  df-so 5602  df-fr 5645  df-se 5646  df-we 5647  df-xp 5699  df-rel 5700  df-cnv 5701  df-co 5702  df-dm 5703  df-rn 5704  df-res 5705  df-ima 5706  df-pred 6329  df-ord 6395  df-on 6396  df-lim 6397  df-suc 6398  df-iota 6522  df-fun 6571  df-fn 6572  df-f 6573  df-f1 6574  df-fo 6575  df-f1o 6576  df-fv 6577  df-isom 6578  df-riota 7395  df-ov 7441  df-oprab 7442  df-mpo 7443  df-om 7895  df-1st 8022  df-2nd 8023  df-frecs 8314  df-wrecs 8345  df-recs 8419  df-rdg 8458  df-1o 8514  df-2o 8515  df-oadd 8518  df-omul 8519  df-er 8753  df-map 8876  df-ixp 8946  df-en 8994  df-dom 8995  df-fin 8997  df-fi 9458  df-wdom 9612  df-card 9986  df-acn 9989  df-topgen 17499  df-pt 17500  df-fbas 21388  df-fg 21389  df-top 22925  df-topon 22942  df-bases 22978  df-cld 23052  df-ntr 23053  df-cls 23054  df-nei 23131  df-cmp 23420  df-fil 23879  df-ufil 23934  df-ufl 23935  df-flim 23972  df-fcls 23974

This theorem is referenced by:  ptcmp  24091  dfac21  43071