Theorem ptex 13067

Description: Existence of the product topology. (Contributed by Jim Kingdon, 19-Mar-2025.)

Assertion

Ref	Expression
ptex	⊢ (𝐹 ∈ 𝑉 → (∏t‘𝐹) ∈ V)

Proof of Theorem ptex

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

Step	Hyp	Ref	Expression
1		df-pt 13064	. . 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		dmeq 4877	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
3	2	fneq2d 5364	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑔 Fn dom 𝑓 ↔ 𝑔 Fn dom 𝐹))
4		fveq1 5574	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
5	4	eleq2d 2274	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ (𝑔‘𝑦) ∈ (𝐹‘𝑦)))
6	2, 5	raleqbidv 2717	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ ∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ (𝐹‘𝑦)))
7	2	difeq1d 3289	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (dom 𝑓 ∖ 𝑧) = (dom 𝐹 ∖ 𝑧))
8	4	unieqd 3860	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ∪ (𝑓‘𝑦) = ∪ (𝐹‘𝑦))
9	8	eqeq2d 2216	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ((𝑔‘𝑦) = ∪ (𝑓‘𝑦) ↔ (𝑔‘𝑦) = ∪ (𝐹‘𝑦)))
10	7, 9	raleqbidv 2717	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦) ↔ ∀𝑦 ∈ (dom 𝐹 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)))
11	10	rexbidv 2506	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦) ↔ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝐹 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)))
12	3, 6, 11	3anbi123d 1324	. . . . . . 7 ⊢ (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 dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝐹 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦))))
13	2	ixpeq1d 6796	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → X𝑦 ∈ dom 𝑓(𝑔‘𝑦) = X𝑦 ∈ dom 𝐹(𝑔‘𝑦))
14	13	eqeq2d 2216	. . . . . . 7 ⊢ (𝑓 = 𝐹 → (𝑥 = X𝑦 ∈ dom 𝑓(𝑔‘𝑦) ↔ 𝑥 = X𝑦 ∈ dom 𝐹(𝑔‘𝑦)))
15	12, 14	anbi12d 473	. . . . . 6 ⊢ (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 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))))
16	15	exbidv 1847	. . . . 5 ⊢ (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 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))))
17	16	abbidv 2322	. . . 4 ⊢ (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 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))})
18	17	fveq2d 5579	. . 3 ⊢ (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‘{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))}))
19		elex 2782	. . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
20		dmexg 4941	. . . . . . . . . 10 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
21		vex 2774	. . . . . . . . . . . . 13 ⊢ 𝑔 ∈ V
22		vex 2774	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
23	21, 22	fvex 5595	. . . . . . . . . . . 12 ⊢ (𝑔‘𝑦) ∈ V
24	23	a1i 9	. . . . . . . . . . 11 ⊢ (𝐹 ∈ 𝑉 → (𝑔‘𝑦) ∈ V)
25	24	ralrimivw 2579	. . . . . . . . . 10 ⊢ (𝐹 ∈ 𝑉 → ∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ V)
26		ixpexgg 6808	. . . . . . . . . 10 ⊢ ((dom 𝐹 ∈ V ∧ ∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ V) → X𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ V)
27	20, 25, 26	syl2anc 411	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝑉 → X𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ V)
28	27	ralrimivw 2579	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑉 → ∀𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)X𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ V)
29		dfiun2g 3958	. . . . . . . 8 ⊢ (∀𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)X𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ V → ∪ 𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)X𝑦 ∈ dom 𝐹(𝑔‘𝑦) = ∪ {𝑥 ∣ ∃𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)𝑥 = X𝑦 ∈ dom 𝐹(𝑔‘𝑦)})
30	28, 29	syl 14	. . . . . . 7 ⊢ (𝐹 ∈ 𝑉 → ∪ 𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)X𝑦 ∈ dom 𝐹(𝑔‘𝑦) = ∪ {𝑥 ∣ ∃𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)𝑥 = X𝑦 ∈ dom 𝐹(𝑔‘𝑦)})
31		rnexg 4942	. . . . . . . . . 10 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V)
32	31	uniexd 4486	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝑉 → ∪ ran 𝐹 ∈ V)
33		mapvalg 6744	. . . . . . . . . 10 ⊢ ((∪ ran 𝐹 ∈ V ∧ dom 𝐹 ∈ V) → (∪ ran 𝐹 ↑𝑚 dom 𝐹) = {𝑔 ∣ 𝑔:dom 𝐹⟶∪ ran 𝐹})
34		mapex 6740	. . . . . . . . . . 11 ⊢ ((dom 𝐹 ∈ V ∧ ∪ ran 𝐹 ∈ V) → {𝑔 ∣ 𝑔:dom 𝐹⟶∪ ran 𝐹} ∈ V)
35	34	ancoms 268	. . . . . . . . . 10 ⊢ ((∪ ran 𝐹 ∈ V ∧ dom 𝐹 ∈ V) → {𝑔 ∣ 𝑔:dom 𝐹⟶∪ ran 𝐹} ∈ V)
36	33, 35	eqeltrd 2281	. . . . . . . . 9 ⊢ ((∪ ran 𝐹 ∈ V ∧ dom 𝐹 ∈ V) → (∪ ran 𝐹 ↑𝑚 dom 𝐹) ∈ V)
37	32, 20, 36	syl2anc 411	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑉 → (∪ ran 𝐹 ↑𝑚 dom 𝐹) ∈ V)
38		iunexg 6203	. . . . . . . 8 ⊢ (((∪ ran 𝐹 ↑𝑚 dom 𝐹) ∈ V ∧ ∀𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)X𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ V) → ∪ 𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)X𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ V)
39	37, 28, 38	syl2anc 411	. . . . . . 7 ⊢ (𝐹 ∈ 𝑉 → ∪ 𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)X𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ V)
40	30, 39	eqeltrrd 2282	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → ∪ {𝑥 ∣ ∃𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)𝑥 = X𝑦 ∈ dom 𝐹(𝑔‘𝑦)} ∈ V)
41		uniexb 4519	. . . . . 6 ⊢ ({𝑥 ∣ ∃𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)𝑥 = X𝑦 ∈ dom 𝐹(𝑔‘𝑦)} ∈ V ↔ ∪ {𝑥 ∣ ∃𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)𝑥 = X𝑦 ∈ dom 𝐹(𝑔‘𝑦)} ∈ V)
42	40, 41	sylibr 134	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → {𝑥 ∣ ∃𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)𝑥 = X𝑦 ∈ dom 𝐹(𝑔‘𝑦)} ∈ V)
43		simp1 999	. . . . . . . . . . 11 ⊢ ((𝑔 Fn dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝐹 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) → 𝑔 Fn dom 𝐹)
44		fvssunirng 5590	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ V → (𝐹‘𝑦) ⊆ ∪ ran 𝐹)
45	44	elv 2775	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑦) ⊆ ∪ ran 𝐹
46	45	sseli 3188	. . . . . . . . . . . . 13 ⊢ ((𝑔‘𝑦) ∈ (𝐹‘𝑦) → (𝑔‘𝑦) ∈ ∪ ran 𝐹)
47	46	ralimi 2568	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ (𝐹‘𝑦) → ∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ ∪ ran 𝐹)
48	47	3ad2ant2 1021	. . . . . . . . . . 11 ⊢ ((𝑔 Fn dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝐹 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) → ∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ ∪ ran 𝐹)
49		ffnfv 5737	. . . . . . . . . . 11 ⊢ (𝑔:dom 𝐹⟶∪ ran 𝐹 ↔ (𝑔 Fn dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ ∪ ran 𝐹))
50	43, 48, 49	sylanbrc 417	. . . . . . . . . 10 ⊢ ((𝑔 Fn dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝐹 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) → 𝑔:dom 𝐹⟶∪ ran 𝐹)
51	32, 20	elmapd 6748	. . . . . . . . . 10 ⊢ (𝐹 ∈ 𝑉 → (𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹) ↔ 𝑔:dom 𝐹⟶∪ ran 𝐹))
52	50, 51	imbitrrid 156	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝑉 → ((𝑔 Fn dom 𝐹 ∧ ∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ (𝐹‘𝑦) ∧ ∃𝑧 ∈ Fin ∀𝑦 ∈ (dom 𝐹 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)) → 𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)))
53	52	anim1d 336	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑉 → (((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 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹) ∧ 𝑥 = X𝑦 ∈ dom 𝐹(𝑔‘𝑦))))
54	53	eximdv 1902	. . . . . . 7 ⊢ (𝐹 ∈ 𝑉 → (∃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 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹) ∧ 𝑥 = X𝑦 ∈ dom 𝐹(𝑔‘𝑦))))
55		df-rex 2489	. . . . . . 7 ⊢ (∃𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)𝑥 = X𝑦 ∈ dom 𝐹(𝑔‘𝑦) ↔ ∃𝑔(𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹) ∧ 𝑥 = X𝑦 ∈ dom 𝐹(𝑔‘𝑦)))
56	54, 55	imbitrrdi 162	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → (∃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 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)𝑥 = X𝑦 ∈ dom 𝐹(𝑔‘𝑦)))
57	56	ss2abdv 3265	. . . . 5 ⊢ (𝐹 ∈ 𝑉 → {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 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)𝑥 = X𝑦 ∈ dom 𝐹(𝑔‘𝑦)})
58	42, 57	ssexd 4183	. . . 4 ⊢ (𝐹 ∈ 𝑉 → {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))} ∈ V)
59		tgvalex 13066	. . . 4 ⊢ ({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))} ∈ 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))}) ∈ V)
60	58, 59	syl 14	. . 3 ⊢ (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))}) ∈ V)
61	1, 18, 19, 60	fvmptd3 5672	. 2 ⊢ (𝐹 ∈ 𝑉 → (∏t‘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))}))
62	61, 60	eqeltrd 2281	1 ⊢ (𝐹 ∈ 𝑉 → (∏t‘𝐹) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 980   = wceq 1372  ∃wex 1514   ∈ wcel 2175  {cab 2190  ∀wral 2483  ∃wrex 2484  Vcvv 2771   ∖ cdif 3162   ⊆ wss 3165  ∪ cuni 3849  ∪ ciun 3926  dom cdm 4674  ran crn 4675   Fn wfn 5265  ⟶wf 5266  ‘cfv 5270  (class class class)co 5943   ↑_𝑚 cmap 6734  Xcixp 6784  Fincfn 6826  topGenctg 13057  ∏_tcpt 13058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-ov 5946  df-oprab 5947  df-mpo 5948  df-map 6736  df-ixp 6785  df-topgen 13063  df-pt 13064

This theorem is referenced by:  prdsex  13072  prdsval  13076  prdsbaslemss  13077  psrval  14399  fnpsr  14400  psrbasg  14407  psrplusgg  14411