Theorem ptval 23758

Description: The value of the product topology function. (Contributed by Mario Carneiro, 3-Feb-2015.)

Hypothesis

Ref	Expression
ptval.1	⊢ 𝐵 = {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))}

Assertion

Ref	Expression
ptval	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → (∏t‘𝐹) = (topGen‘𝐵))

Distinct variable groups:   𝑥,𝑔,𝑦,𝑧,𝐴   𝑔,𝐹,𝑥,𝑦,𝑧   𝑔,𝑉,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐵(𝑥, 𝑦, 𝑧, 𝑔)

Proof of Theorem ptval

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-pt 17515	. 2 ⊢ ∏t = (𝑓 ∈ V ↦ (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔‘𝑦))}))
2		simpr 490	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
3	2	dmeqd 5897	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝑓 = dom 𝐹)
4		fndm 6642	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	4	ad2antlr 740	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝐹 = 𝐴)
6	3, 5	eqtrd 2800	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝑓 = 𝐴)
7	6	fneq2d 6633	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑔 Fn dom 𝑓 ↔ 𝑔 Fn 𝐴))
8	2	fveq1d 6887	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑓‘𝑦) = (𝐹‘𝑦))
9	8	eleq2d 2851	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ (𝑔‘𝑦) ∈ (𝐹‘𝑦)))
10	6, 9	raleqbidv 3340	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦)))
11	6	difeq1d 4080	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (dom 𝑓 ∖ 𝑧) = (𝐴 ∖ 𝑧))
12	8	unieqd 4887	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ∪ (𝑓‘𝑦) = ∪ (𝐹‘𝑦))
13	12	eqeq2d 2776	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔‘𝑦) = ∪ (𝑓‘𝑦) ↔ (𝑔‘𝑦) = ∪ (𝐹‘𝑦)))
14	11, 13	raleqbidv 3340	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)))
15	14	rexbidv 3191	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦) ↔ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)))
16	7, 10, 15	3anbi123d 1464	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦)) ↔ (𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))))
17	6	ixpeq1d 8909	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → X𝑦 ∈ dom 𝑓(𝑔‘𝑦) = X𝑦 ∈ 𝐴 (𝑔‘𝑦))
18	17	eqeq2d 2776	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑥 = X𝑦 ∈ dom 𝑓(𝑔‘𝑦) ↔ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
19	16, 18	anbi12d 644	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔‘𝑦)) ↔ ((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))))
20	19	exbidv 1954	. . . . 5 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔‘𝑦)) ↔ ∃𝑔((𝑔 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) ∧ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦))))
21	20	abbidv 2831	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → {𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(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))})
22		ptval.1	. . . 4 ⊢ 𝐵 = {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))}
23	21, 22	eqtr4di 2818	. . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → {𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔‘𝑦))} = 𝐵)
24	23	fveq2d 6889	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (topGen‘{𝑥 ∣ ∃𝑔((𝑔 Fn dom 𝑓 ∧ ∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦)) ∧ 𝑥 = X𝑦 ∈ dom 𝑓(𝑔‘𝑦))}) = (topGen‘𝐵))
25		fnex 7219	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
26	25	ancoms 464	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → 𝐹 ∈ V)
27		fvexd 6900	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → (topGen‘𝐵) ∈ V)
28	1, 24, 26, 27	fvmptd2 7002	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → (∏t‘𝐹) = (topGen‘𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ∖ cdif 3903  ∪ cuni 4874  dom cdm 5663   Fn wfn 6535  ‘cfv 6540  Xcixp 8897  Fincfn 8945  topGenctg 17508  ∏_tcpt 17509

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ixp 8898  df-pt 17515

This theorem is used by:  pttop  23770  ptopn  23771  ptuni  23782  ptval2  23789  ptpjcn  23799  ptpjopn  23800  ptclsg  23803  ptcnp  23810  prdstopn  23816  xkoptsub  23842  ptcmplem1  24240  tmdgsum2  24284  prdsxmslem2  24717  ptrecube  38304