Theorem ptval2 23975

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 6739	. . 3 ⊢ (𝐹:𝐴⟶Top → 𝐹 Fn 𝐴)
2		ptval2.1	. . . 4 ⊢ 𝐽 = (∏t‘𝐹)
3		eqid 2763	. . . . 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 23944	. . . 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 2810	. . 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 605	. 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 2763	. . . . 5 ⊢ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)
8	3, 7	ptbasfi 23955	. . . 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 23968	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = ∪ 𝐽)
10		ptval2.2	. . . . . . . 8 ⊢ 𝑋 = ∪ 𝐽
11	9, 10	eqtr4di 2816	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = 𝑋)
12	11	sneqd 4596	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → {X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛)} = {𝑋})
13	11	3ad2ant1 1151	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) ∧ 𝑘 ∈ 𝐴 ∧ 𝑢 ∈ (𝐹‘𝑘)) → X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) = 𝑋)
14	13	mpteq1d 5201	. . . . . . . . . . 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 5865	. . . . . . . . . 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 6069	. . . . . . . . 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 7528	. . . . . . . 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 2816	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → (𝑘 ∈ 𝐴, 𝑢 ∈ (𝐹‘𝑘) ↦ (◡(𝑤 ∈ X𝑛 ∈ 𝐴 ∪ (𝐹‘𝑛) ↦ (𝑤‘𝑘)) “ 𝑢)) = 𝐺)
20	19	rneqd 5932	. . . . . 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 4171	. . . . 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 6920	. . . 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 2798	. . 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 6920	. 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 2798	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶Top) → 𝐽 = (topGen‘(fi‘({𝑋} ∪ ran 𝐺))))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090   ∖ cdif 4048   ∪ cun 4152  {csn 4588  ∪ cuni 4872   ↦ cmpt 5192  ^◡ccnv 5657  ran crn 5659   “ cima 5661   Fn wfn 6561  ⟶wf 6562  ‘cfv 6566   ∈ cmpo 7453  Xcixp 8963  Fincfn 9011  ficfi 9439  topGenctg 17658  ∏_tcpt 17659  Topctop 23266

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-int 4913  df-iun 4958  df-iin 4959  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-lim 6393  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-om 7912  df-1st 8035  df-2nd 8036  df-1o 8514  df-2o 8515  df-ixp 8964  df-en 9012  df-dom 9013  df-fin 9015  df-fi 9440  df-topgen 17664  df-pt 17665  df-top 23267  df-bases 23319

This theorem is used by:  ptrescn  24013  ptrest  38805