Theorem ptval 21373

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 16105	. . 3 ⊢ ∏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	1	a1i 11	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → ∏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))})))
3		simpr 477	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
4	3	dmeqd 5326	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝑓 = dom 𝐹)
5		fndm 5990	. . . . . . . . . . 11 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴)
6	5	ad2antlr 763	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝐹 = 𝐴)
7	4, 6	eqtrd 2656	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → dom 𝑓 = 𝐴)
8	7	fneq2d 5982	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑔 Fn dom 𝑓 ↔ 𝑔 Fn 𝐴))
9	3	fveq1d 6193	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑓‘𝑦) = (𝐹‘𝑦))
10	9	eleq2d 2687	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ (𝑔‘𝑦) ∈ (𝐹‘𝑦)))
11	7, 10	raleqbidv 3152	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝑔‘𝑦) ∈ (𝐹‘𝑦)))
12	7	difeq1d 3727	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (dom 𝑓 ∖ 𝑧) = (𝐴 ∖ 𝑧))
13	9	unieqd 4446	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ∪ (𝑓‘𝑦) = ∪ (𝐹‘𝑦))
14	13	eqeq2d 2632	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → ((𝑔‘𝑦) = ∪ (𝑓‘𝑦) ↔ (𝑔‘𝑦) = ∪ (𝐹‘𝑦)))
15	12, 14	raleqbidv 3152	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦) ↔ ∀𝑦 ∈ (𝐴 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)))
16	15	rexbidv 3052	. . . . . . . 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)))
17	8, 11, 16	3anbi123d 1399	. . . . . . 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))))
18	7	ixpeq1d 7920	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → X𝑦 ∈ dom 𝑓(𝑔‘𝑦) = X𝑦 ∈ 𝐴 (𝑔‘𝑦))
19	18	eqeq2d 2632	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) ∧ 𝑓 = 𝐹) → (𝑥 = X𝑦 ∈ dom 𝑓(𝑔‘𝑦) ↔ 𝑥 = X𝑦 ∈ 𝐴 (𝑔‘𝑦)))
20	17, 19	anbi12d 747	. . . . . 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))))
21	20	exbidv 1850	. . . . 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))))
22	21	abbidv 2741	. . . 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))})
23		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))}
24	22, 23	syl6eqr 2674	. . 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)
25	24	fveq2d 6195	. 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))
26		fnex 6481	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V)
27	26	ancoms 469	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → 𝐹 ∈ V)
28		fvexd 6203	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → (topGen‘𝐵) ∈ V)
29	2, 25, 27, 28	fvmptd 6288	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹 Fn 𝐴) → (∏t‘𝐹) = (topGen‘𝐵))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ∖ cdif 3571  ∪ cuni 4436   ↦ cmpt 4729  dom cdm 5114   Fn wfn 5883  ‘cfv 5888  Xcixp 7908  Fincfn 7955  topGenctg 16098  ∏_tcpt 16099

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ixp 7909  df-pt 16105

This theorem is referenced by:  pttop  21385  ptopn  21386  ptuni  21397  ptval2  21404  ptpjcn  21414  ptpjopn  21415  ptclsg  21418  ptcnp  21425  prdstopn  21431  xkoptsub  21457  ptcmplem1  21856  tmdgsum2  21900  prdsxmslem2  22334  ptrecube  33409