Theorem ptcmpg 24369

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 24370). (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 2923	. . . 4 ⊢ Ⅎ𝑘(𝐹‘𝑎)
3		nfcv 2923	. . . 4 ⊢ Ⅎ𝑎(𝐹‘𝑘)
4		nfcv 2923	. . . 4 ⊢ Ⅎ𝑘(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏)
5		nfcv 2923	. . . 4 ⊢ Ⅎ𝑢(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏)
6		nfcv 2923	. . . 4 ⊢ Ⅎ𝑎(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)
7		nfcv 2923	. . . 4 ⊢ Ⅎ𝑏(◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)
8		fveq2 6883	. . . 4 ⊢ (𝑎 = 𝑘 → (𝐹‘𝑎) = (𝐹‘𝑘))
9		fveq2 6883	. . . . . . . 8 ⊢ (𝑎 = 𝑘 → (𝑤‘𝑎) = (𝑤‘𝑘))
10	9	mpteq2dv 5199	. . . . . . 7 ⊢ (𝑎 = 𝑘 → (𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) = (𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)))
11	10	cnveqd 5853	. . . . . 6 ⊢ (𝑎 = 𝑘 → ◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) = ◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)))
12	11	imaeq1d 6051	. . . . 5 ⊢ (𝑎 = 𝑘 → (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑎)) “ 𝑏) = (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑏))
13		imaeq2 6048	. . . . 5 ⊢ (𝑏 = 𝑢 → (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑏) = (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢))
14	12, 13	sylan9eq 2816	. . . 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 7511	. . 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 6883	. . . . 5 ⊢ (𝑛 = 𝑚 → (𝐹‘𝑛) = (𝐹‘𝑚))
17	16	unieqd 4880	. . . 4 ⊢ (𝑛 = 𝑚 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑚))
18	17	cbvixpv 8936	. . 3 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = X𝑚 ∈ 𝐴 ∪ (𝐹‘𝑚)
19		simp1 1154	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐴 ∈ 𝑉)
20		simp2 1155	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐹:𝐴⟶Comp)
21		cmptop 23706	. . . . . . . 8 ⊢ (𝑘 ∈ Comp → 𝑘 ∈ Top)
22	21	ssriv 3935	. . . . . . 7 ⊢ Comp ⊆ Top
23		fss 6724	. . . . . . 7 ⊢ ((𝐹:𝐴⟶Comp ∧ Comp ⊆ Top) → 𝐹:𝐴⟶Top)
24	20, 22, 23	sylancl 598	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐹:𝐴⟶Top)
25	1	ptuni 23906	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ 𝐽)
26	19, 24, 25	syl2anc 596	. . . . 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 2814	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = 𝑋)
29		simp3 1156	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝑋 ∈ (UFL ∩ dom card))
30	28, 29	eqeltrd 2861	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ∈ (UFL ∩ dom card))
31	15, 18, 19, 20, 30	ptcmplem5 24368	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → (∏t‘𝐹) ∈ Comp)
32	1, 31	eqeltrid 2865	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Comp ∧ 𝑋 ∈ (UFL ∩ dom card)) → 𝐽 ∈ Comp)

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145   ∩ cin 3898   ⊆ wss 3899  ∪ cuni 4867   ↦ cmpt 5186  ^◡ccnv 5650  dom cdm 5651   “ cima 5654  ⟶wf 6533  ‘cfv 6537   ∈ cmpo 7420  Xcixp 8918  cardccrd 10009  ∏_tcpt 17602  Topctop 23204  Compccmp 23697  UFLcufl 24212

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 7749

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 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-oadd 8473  df-omul 8474  df-er 8710  df-map 8842  df-ixp 8919  df-en 8967  df-dom 8968  df-fin 8970  df-fi 9396  df-wdom 9552  df-card 10013  df-acn 10016  df-topgen 17607  df-pt 17608  df-fbas 21668  df-fg 21669  df-top 23205  df-topon 23222  df-bases 23257  df-cld 23330  df-ntr 23331  df-cls 23332  df-nei 23409  df-cmp 23698  df-fil 24158  df-ufil 24213  df-ufl 24214  df-flim 24251  df-fcls 24253

This theorem is used by:  ptcmp  24370  dfac21  44052