Theorem ptval2 23456

Description: The value of the product topology function. (Contributed by Mario Carneiro, 7-Feb-2015.)

Hypotheses

Ref	Expression
ptval2.1	⊢ 𝐽 = (∏t‘𝐹)
ptval2.2	⊢ 𝑋 = ∪ 𝐽
ptval2.3	⊢ 𝐺 = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))

Assertion

Ref	Expression
ptval2	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐽 = (topGen‘(fi‘({𝑋} ∪ ran 𝐺))))

Distinct variable groups:   𝑢,𝑘,𝑤,𝐴   𝑘,𝐹,𝑢,𝑤   𝑘,𝑉,𝑢,𝑤   𝑤,𝑋

Allowed substitution hints:   𝐺(𝑤,𝑢,𝑘)   𝐽(𝑤,𝑢,𝑘)   𝑋(𝑢,𝑘)

Proof of Theorem ptval2

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

Step	Hyp	Ref	Expression
1		ffn 6710	. . 3 ⊢ (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
2		ptval2.1	. . . 4 ⊢ 𝐽 = (∏t‘𝐹)
3		eqid 2726	. . . . 5 ⊢ {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} = {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}
4	3	ptval 23425	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → (∏t‘𝐹) = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
5	2, 4	eqtrid 2778	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → 𝐽 = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
6	1, 5	sylan2 592	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐽 = (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}))
7		eqid 2726	. . . . 5 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
8	3, 7	ptbasfi 23436	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} = (fi‘({X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)))))
9	2	ptuni 23449	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ 𝐽)
10		ptval2.2	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
11	9, 10	eqtr4di 2784	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = 𝑋)
12	11	sneqd 4635	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)} = {𝑋})
13	11	3ad2ant1 1130	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ 𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘)) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = 𝑋)
14	13	mpteq1d 5236	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ 𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘)) → (𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) = (𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)))
15	14	cnveqd 5868	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ 𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘)) → ◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) = ◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)))
16	15	imaeq1d 6051	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ 𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘)) → (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢) = (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
17	16	mpoeq3dva 7481	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)) = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢)))
18		ptval2.3	. . . . . . . 8 ⊢ 𝐺 = (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ 𝑋 ↦ (𝑤‘𝑘)) “ 𝑢))
19	17, 18	eqtr4di 2784	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)) = 𝐺)
20	19	rneqd 5930	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)) = ran 𝐺)
21	12, 20	uneq12d 4159	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → ({X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢))) = ({𝑋} ∪ ran 𝐺))
22	21	fveq2d 6888	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (fi‘({X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)} ∪ ran (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)))) = (fi‘({𝑋} ∪ ran 𝐺)))
23	8, 22	eqtrd 2766	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))} = (fi‘({𝑋} ∪ ran 𝐺)))
24	23	fveq2d 6888	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))}) = (topGen‘(fi‘({𝑋} ∪ ran 𝐺))))
25	6, 24	eqtrd 2766	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐽 = (topGen‘(fi‘({𝑋} ∪ ran 𝐺))))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2703  ∀wral 3055  ∃wrex 3064   ∖ cdif 3940   ∪ cun 3941  {csn 4623  ∪ cuni 4902   ↦ cmpt 5224  ^◡ccnv 5668  ran crn 5670   “ cima 5672   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536   ∈ cmpo 7406  Xcixp 8890  Fincfn 8938  ficfi 9404  topGenctg 17390  ∏_tcpt 17391  Topctop 22746

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-1o 8464  df-er 8702  df-ixp 8891  df-en 8939  df-dom 8940  df-fin 8942  df-fi 9405  df-topgen 17396  df-pt 17397  df-top 22747  df-bases 22800

This theorem is referenced by:  ptrescn  23494  ptrest  36998