Theorem ptex 13410

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 13407	. . 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 4937	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
3	2	fneq2d 5428	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (𝑔 Fn dom 𝑓 ↔ 𝑔 Fn dom 𝐹))
4		fveq1 5647	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑦) = (𝐹‘𝑦))
5	4	eleq2d 2301	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → ((𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ (𝑔‘𝑦) ∈ (𝐹‘𝑦)))
6	2, 5	raleqbidv 2747	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ dom 𝑓(𝑔‘𝑦) ∈ (𝑓‘𝑦) ↔ ∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ (𝐹‘𝑦)))
7	2	difeq1d 3326	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → (dom 𝑓 ∖ 𝑧) = (dom 𝐹 ∖ 𝑧))
8	4	unieqd 3909	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → ∪ (𝑓‘𝑦) = ∪ (𝐹‘𝑦))
9	8	eqeq2d 2243	. . . . . . . . . 10 ⊢ (𝑓 = 𝐹 → ((𝑔‘𝑦) = ∪ (𝑓‘𝑦) ↔ (𝑔‘𝑦) = ∪ (𝐹‘𝑦)))
10	7, 9	raleqbidv 2747	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (∀𝑦 ∈ (dom 𝑓 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝑓‘𝑦) ↔ ∀𝑦 ∈ (dom 𝐹 ∖ 𝑧)(𝑔‘𝑦) = ∪ (𝐹‘𝑦)))
11	10	rexbidv 2534	. . . . . . . 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 1349	. . . . . . 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 6922	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → X𝑦 ∈ dom 𝑓(𝑔‘𝑦) = X𝑦 ∈ dom 𝐹(𝑔‘𝑦))
14	13	eqeq2d 2243	. . . . . . 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 1873	. . . . 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 2350	. . . 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 5652	. . 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 2815	. . 3 ⊢ (𝐹 ∈ 𝑉 → 𝐹 ∈ V)
20		dmexg 5002	. . . . . . . . . 10 ⊢ (𝐹 ∈ 𝑉 → dom 𝐹 ∈ V)
21		vex 2806	. . . . . . . . . . . . 13 ⊢ 𝑔 ∈ V
22		vex 2806	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
23	21, 22	fvex 5668	. . . . . . . . . . . 12 ⊢ (𝑔‘𝑦) ∈ V
24	23	a1i 9	. . . . . . . . . . 11 ⊢ (𝐹 ∈ 𝑉 → (𝑔‘𝑦) ∈ V)
25	24	ralrimivw 2607	. . . . . . . . . 10 ⊢ (𝐹 ∈ 𝑉 → ∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ V)
26		ixpexgg 6934	. . . . . . . . . 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 2607	. . . . . . . 8 ⊢ (𝐹 ∈ 𝑉 → ∀𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)X𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ V)
29		dfiun2g 4007	. . . . . . . 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 5003	. . . . . . . . . 10 ⊢ (𝐹 ∈ 𝑉 → ran 𝐹 ∈ V)
32	31	uniexd 4543	. . . . . . . . 9 ⊢ (𝐹 ∈ 𝑉 → ∪ ran 𝐹 ∈ V)
33		mapvalg 6870	. . . . . . . . . 10 ⊢ ((∪ ran 𝐹 ∈ V ∧ dom 𝐹 ∈ V) → (∪ ran 𝐹 ↑𝑚 dom 𝐹) = {𝑔 ∣ 𝑔:dom 𝐹⟶∪ ran 𝐹})
34		mapex 6866	. . . . . . . . . . 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 2308	. . . . . . . . 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 6290	. . . . . . . 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 2309	. . . . . 6 ⊢ (𝐹 ∈ 𝑉 → ∪ {𝑥 ∣ ∃𝑔 ∈ (∪ ran 𝐹 ↑𝑚 dom 𝐹)𝑥 = X𝑦 ∈ dom 𝐹(𝑔‘𝑦)} ∈ V)
41		uniexb 4576	. . . . . 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 1024	. . . . . . . . . . 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 5663	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ V → (𝐹‘𝑦) ⊆ ∪ ran 𝐹)
45	44	elv 2807	. . . . . . . . . . . . . 14 ⊢ (𝐹‘𝑦) ⊆ ∪ ran 𝐹
46	45	sseli 3224	. . . . . . . . . . . . 13 ⊢ ((𝑔‘𝑦) ∈ (𝐹‘𝑦) → (𝑔‘𝑦) ∈ ∪ ran 𝐹)
47	46	ralimi 2596	. . . . . . . . . . . 12 ⊢ (∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ (𝐹‘𝑦) → ∀𝑦 ∈ dom 𝐹(𝑔‘𝑦) ∈ ∪ ran 𝐹)
48	47	3ad2ant2 1046	. . . . . . . . . . 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 5813	. . . . . . . . . . 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 6874	. . . . . . . . . 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 1928	. . . . . . 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 2517	. . . . . . 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 3301	. . . . 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 4234	. . . 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 13409	. . . 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 5749	. 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 2308	1 ⊢ (𝐹 ∈ 𝑉 → (∏t‘𝐹) ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   ∧ w3a 1005   = wceq 1398  ∃wex 1541   ∈ wcel 2202  {cab 2217  ∀wral 2511  ∃wrex 2512  Vcvv 2803   ∖ cdif 3198   ⊆ wss 3201  ∪ cuni 3898  ∪ ciun 3975  dom cdm 4731  ran crn 4732   Fn wfn 5328  ⟶wf 5329  ‘cfv 5333  (class class class)co 6028   ↑_𝑚 cmap 6860  Xcixp 6910  Fincfn 6952  topGenctg 13400  ∏_tcpt 13401

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-map 6862  df-ixp 6911  df-topgen 13406  df-pt 13407

This theorem is referenced by:  prdsex  13415  prdsval  13419  prdsbaslemss  13420  psrval  14745  fnpsr  14746  psrbasg  14758  psrplusgg  14762