Theorem ptval 23578

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 17489	. 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 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
3	2	dmeqd 5916	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝑓 = dom 𝐹)
4		fndm 6671	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
5	4	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝐹 = 𝐴)
6	3, 5	eqtrd 2777	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝑓 = 𝐴)
7	6	fneq2d 6662	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑔 Fn dom 𝑓 ↔ 𝑔 Fn 𝐴))
8	2	fveq1d 6908	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑓‘𝑦) = (𝐹‘𝑦))
9	8	eleq2d 2827	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ (𝑔‘𝑦) ∈ (𝐹‘𝑦)))
10	6, 9	raleqbidv 3346	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦)))
11	6	difeq1d 4125	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (dom 𝑓 ∖ 𝑧) = (𝐴 ∖ 𝑧))
12	8	unieqd 4920	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ∪ (𝑓‘𝑦) = ∪ (𝐹‘𝑦))
13	12	eqeq2d 2748	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔‘𝑦) = ∪ (𝑓‘𝑦) ↔ (𝑔‘𝑦) = ∪ (𝐹‘𝑦)))
14	11, 13	raleqbidv 3346	. . . . . . . . 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 3179	. . . . . . . 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 1438	. . . . . . 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 8949	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → X𝑦 ∈ dom 𝑓(𝑔‘𝑦) = X𝑦 ∈ 𝐴 (𝑔‘𝑦))
18	17	eqeq2d 2748	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑥 = X𝑦 ∈ dom 𝑓(𝑔‘𝑦) ↔ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
19	16, 18	anbi12d 632	. . . . . 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 1921	. . . . 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 2808	. . . 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 2795	. . 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 6910	. 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 7237	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
26	25	ancoms 458	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → 𝐹 ∈ V)
27		fvexd 6921	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → (topGen‘𝐵) ∈ V)
28	1, 24, 26, 27	fvmptd2 7024	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → (∏t‘𝐹) = (topGen‘𝐵))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714  ∀wral 3061  ∃wrex 3070  Vcvv 3480   ∖ cdif 3948  ∪ cuni 4907  dom cdm 5685   Fn wfn 6556  ‘cfv 6561  Xcixp 8937  Fincfn 8985  topGenctg 17482  ∏_tcpt 17483

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ixp 8938  df-pt 17489

This theorem is referenced by:  pttop  23590  ptopn  23591  ptuni  23602  ptval2  23609  ptpjcn  23619  ptpjopn  23620  ptclsg  23623  ptcnp  23630  prdstopn  23636  xkoptsub  23662  ptcmplem1  24060  tmdgsum2  24104  prdsxmslem2  24542  ptrecube  37627