Theorem ptcmpg 23967

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 23968). (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 2894	. . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑎)
3		nfcv 2894	. . . 4 ⊢ Ⅎ𝑎(𝐹‘𝑘)
4		nfcv 2894	. . . 4 ⊢ Ⅎ𝑘(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏)
5		nfcv 2894	. . . 4 ⊢ Ⅎ𝑢(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏)
6		nfcv 2894	. . . 4 ⊢ Ⅎ𝑎(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)
7		nfcv 2894	. . . 4 ⊢ Ⅎ𝑏(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)
8		fveq2 6817	. . . 4 ⊢ (𝑎 = 𝑘 → (𝐹‘𝑎) = (𝐹‘𝑘))
9		fveq2 6817	. . . . . . . 8 ⊢ (𝑎 = 𝑘 → (𝑤‘𝑎) = (𝑤‘𝑘))
10	9	mpteq2dv 5180	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) = (𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)))
11	10	cnveqd 5810	. . . . . 6 ⊢ (𝑎 = 𝑘 → ◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) = ◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)))
12	11	imaeq1d 6003	. . . . 5 ⊢ (𝑎 = 𝑘 → (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏) = (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑏))
13		imaeq2 6000	. . . . 5 ⊢ (𝑏 = 𝑢 → (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑏) = (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢))
14	12, 13	sylan9eq 2786	. . . 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 7434	. . 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 6817	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
17	16	unieqd 4867	. . . 4 ⊢ (𝑛 = 𝑚 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑚))
18	17	cbvixpv 8834	. . 3 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = X𝑚 ∈ 𝐴 ∪ (𝐹‘𝑚)
19		simp1 1136	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐴 ∈ 𝑉)
20		simp2 1137	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐹:𝐴⟶Comp)
21		cmptop 23305	. . . . . . . 8 ⊢ (𝑘 ∈ Comp → 𝑘 ∈ Top)
22	21	ssriv 3933	. . . . . . 7 ⊢ Comp ⊆ Top
23		fss 6662	. . . . . . 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 23504	. . . . . 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 2784	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = 𝑋)
29		simp3 1138	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝑋 ∈ (UFL ∩ dom card))
30	28, 29	eqeltrd 2831	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ (UFL ∩ dom card))
31	15, 18, 19, 20, 30	ptcmplem5 23966	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → (∏t‘𝐹) ∈ Comp)
32	1, 31	eqeltrid 2835	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐽 ∈ Comp)

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ∩ cin 3896   ⊆ wss 3897  ∪ cuni 4854   ↦ cmpt 5167  ^◡ccnv 5610  dom cdm 5611   “ cima 5614  ⟶wf 6472  ‘cfv 6476   ∈ cmpo 7343  Xcixp 8816  cardccrd 9823  ∏_tcpt 17337  Topctop 22803  Compccmp 23296  UFLcufl 23810

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-int 4893  df-iun 4938  df-iin 4939  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5506  df-eprel 5511  df-po 5519  df-so 5520  df-fr 5564  df-se 5565  df-we 5566  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-pred 6243  df-ord 6304  df-on 6305  df-lim 6306  df-suc 6307  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-isom 6485  df-riota 7298  df-ov 7344  df-oprab 7345  df-mpo 7346  df-om 7792  df-1st 7916  df-2nd 7917  df-frecs 8206  df-wrecs 8237  df-recs 8286  df-rdg 8324  df-1o 8380  df-2o 8381  df-oadd 8384  df-omul 8385  df-er 8617  df-map 8747  df-ixp 8817  df-en 8865  df-dom 8866  df-fin 8868  df-fi 9290  df-wdom 9446  df-card 9827  df-acn 9830  df-topgen 17342  df-pt 17343  df-fbas 21283  df-fg 21284  df-top 22804  df-topon 22821  df-bases 22856  df-cld 22929  df-ntr 22930  df-cls 22931  df-nei 23008  df-cmp 23297  df-fil 23756  df-ufil 23811  df-ufl 23812  df-flim 23849  df-fcls 23851

This theorem is referenced by:  ptcmp  23968  dfac21  43099