Theorem comppfsc 23036

Description: A space where every open cover has a point-finite subcover is compact. This is significant in part because it shows half of the proposition that if only half the generalization in the definition of metacompactness (and consequently paracompactness) is performed, one does not obtain any more spaces. (Contributed by Jeff Hankins, 21-Jan-2010.) (Proof shortened by Mario Carneiro, 11-Sep-2015.)

Hypothesis

Ref	Expression
comppfsc.1	⊢ 𝑋 = ∪ 𝐽

Assertion

Ref	Expression
comppfsc	⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))))

Distinct variable groups:   𝑐,𝑑,𝐽   𝑋,𝑐,𝑑

Proof of Theorem comppfsc

Dummy variables 𝑎 𝑏 𝑓 𝑝 𝑞 𝑠 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpwi 4610	. . . 4 ⊢ (𝑐 ∈ 𝒫 𝐽 → 𝑐 ⊆ 𝐽)
2		comppfsc.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	cmpcov 22893	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)
4		elfpw 9354	. . . . . . . 8 ⊢ (𝑑 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑑 ⊆ 𝑐 ∧ 𝑑 ∈ Fin))
5		finptfin 23022	. . . . . . . . . . 11 ⊢ (𝑑 ∈ Fin → 𝑑 ∈ PtFin)
6	5	anim1i 616	. . . . . . . . . 10 ⊢ ((𝑑 ∈ Fin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))
7	6	anassrs 469	. . . . . . . . 9 ⊢ (((𝑑 ∈ Fin ∧ 𝑑 ⊆ 𝑐) ∧ 𝑋 = ∪ 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))
8	7	ancom1s 652	. . . . . . . 8 ⊢ (((𝑑 ⊆ 𝑐 ∧ 𝑑 ∈ Fin) ∧ 𝑋 = ∪ 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))
9	4, 8	sylanb 582	. . . . . . 7 ⊢ ((𝑑 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = ∪ 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))
10	9	reximi2 3080	. . . . . 6 ⊢ (∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))
11	3, 10	syl 17	. . . . 5 ⊢ ((𝐽 ∈ Comp ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))
12	11	3exp 1120	. . . 4 ⊢ (𝐽 ∈ Comp → (𝑐 ⊆ 𝐽 → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))))
13	1, 12	syl5 34	. . 3 ⊢ (𝐽 ∈ Comp → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))))
14	13	ralrimiv 3146	. 2 ⊢ (𝐽 ∈ Comp → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))
15		elpwi 4610	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝐽 → 𝑎 ⊆ 𝐽)
16		0elpw 5355	. . . . . . . . . . 11 ⊢ ∅ ∈ 𝒫 𝑎
17		0fin 9171	. . . . . . . . . . 11 ⊢ ∅ ∈ Fin
18	16, 17	elini 4194	. . . . . . . . . 10 ⊢ ∅ ∈ (𝒫 𝑎 ∩ Fin)
19		unieq 4920	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → ∪ 𝑏 = ∪ ∅)
20		uni0 4940	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
21	19, 20	eqtrdi 2789	. . . . . . . . . . 11 ⊢ (𝑏 = ∅ → ∪ 𝑏 = ∅)
22	21	rspceeqv 3634	. . . . . . . . . 10 ⊢ ((∅ ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = ∅) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)
23	18, 22	mpan 689	. . . . . . . . 9 ⊢ (𝑋 = ∅ → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)
24	23	a1i13 27	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑋 = ∅ → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))
25		n0 4347	. . . . . . . . 9 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑋)
26		simp2 1138	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → 𝑋 = ∪ 𝑎)
27	26	eleq2d 2820	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝑎))
28	27	biimpd 228	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑥 ∈ 𝑋 → 𝑥 ∈ ∪ 𝑎))
29		eluni2 4913	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑎 ↔ ∃𝑠 ∈ 𝑎 𝑥 ∈ 𝑠)
30	28, 29	imbitrdi 250	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑥 ∈ 𝑋 → ∃𝑠 ∈ 𝑎 𝑥 ∈ 𝑠))
31		simpl3 1194	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑎 ⊆ 𝐽)
32		simprl 770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ∈ 𝑎)
33	31, 32	sseldd 3984	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ∈ 𝐽)
34		elssuni 4942	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ 𝐽 → 𝑠 ⊆ ∪ 𝐽)
35	34, 2	sseqtrrdi 4034	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑠 ∈ 𝐽 → 𝑠 ⊆ 𝑋)
36	33, 35	syl 17	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ⊆ 𝑋)
37	36	ralrimivw 3151	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∀𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋)
38		iunss 5049	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋 ↔ ∀𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋)
39	37, 38	sylibr 233	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∪ 𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋)
40		ssequn1 4181	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋 ↔ (∪ 𝑝 ∈ 𝑎 𝑠 ∪ 𝑋) = 𝑋)
41	39, 40	sylib 217	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (∪ 𝑝 ∈ 𝑎 𝑠 ∪ 𝑋) = 𝑋)
42		simpl2 1193	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = ∪ 𝑎)
43		uniiun 5062	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑎 = ∪ 𝑝 ∈ 𝑎 𝑝
44	42, 43	eqtrdi 2789	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = ∪ 𝑝 ∈ 𝑎 𝑝)
45	44	uneq2d 4164	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (∪ 𝑝 ∈ 𝑎 𝑠 ∪ 𝑋) = (∪ 𝑝 ∈ 𝑎 𝑠 ∪ ∪ 𝑝 ∈ 𝑎 𝑝))
46	41, 45	eqtr3d 2775	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = (∪ 𝑝 ∈ 𝑎 𝑠 ∪ ∪ 𝑝 ∈ 𝑎 𝑝))
47		iunun 5097	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑝 ∈ 𝑎 (𝑠 ∪ 𝑝) = (∪ 𝑝 ∈ 𝑎 𝑠 ∪ ∪ 𝑝 ∈ 𝑎 𝑝)
48		vex 3479	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑠 ∈ V
49		vex 3479	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑝 ∈ V
50	48, 49	unex 7733	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∪ 𝑝) ∈ V
51	50	dfiun3 5966	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑝 ∈ 𝑎 (𝑠 ∪ 𝑝) = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))
52	47, 51	eqtr3i 2763	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑝 ∈ 𝑎 𝑠 ∪ ∪ 𝑝 ∈ 𝑎 𝑝) = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))
53	46, 52	eqtrdi 2789	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)))
54		simpll1 1213	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → 𝐽 ∈ Top)
55	33	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → 𝑠 ∈ 𝐽)
56	31	sselda 3983	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → 𝑝 ∈ 𝐽)
57		unopn 22405	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) → (𝑠 ∪ 𝑝) ∈ 𝐽)
58	54, 55, 56, 57	syl3anc 1372	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → (𝑠 ∪ 𝑝) ∈ 𝐽)
59	58	fmpttd 7115	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)):𝑎⟶𝐽)
60	59	frnd 6726	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽)
61		elpw2g 5345	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 ∈ Top → (ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽))
62	61	3ad2ant1 1134	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽))
63	62	adantr 482	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽))
64	60, 63	mpbird 257	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽)
65		unieq 4920	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ∪ 𝑐 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)))
66	65	eqeq2d 2744	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → (𝑋 = ∪ 𝑐 ↔ 𝑋 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))))
67		sseq2 4009	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → (𝑑 ⊆ 𝑐 ↔ 𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))))
68	67	anbi1d 631	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ((𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑) ↔ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)))
69	68	rexbidv 3179	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → (∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑) ↔ ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)))
70	66, 69	imbi12d 345	. . . . . . . . . . . . . . . 16 ⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ((𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) ↔ (𝑋 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑))))
71	70	rspcv 3609	. . . . . . . . . . . . . . 15 ⊢ (ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → (𝑋 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑))))
72	64, 71	syl 17	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → (𝑋 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑))))
73	53, 72	mpid 44	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)))
74		simprr 772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ 𝑠)
75		ssel2 3978	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ⊆ 𝐽 ∧ 𝑠 ∈ 𝑎) → 𝑠 ∈ 𝐽)
76	75	3ad2antl3 1188	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑠 ∈ 𝑎) → 𝑠 ∈ 𝐽)
77	76	adantrr 716	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ∈ 𝐽)
78		elunii 4914	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 ∈ 𝐽) → 𝑥 ∈ ∪ 𝐽)
79	74, 77, 78	syl2anc 585	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ ∪ 𝐽)
80	79, 2	eleqtrrdi 2845	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ 𝑋)
81	80	adantr 482	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑥 ∈ 𝑋)
82		simprr 772	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑋 = ∪ 𝑑)
83	81, 82	eleqtrd 2836	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑥 ∈ ∪ 𝑑)
84		eqid 2733	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑑 = ∪ 𝑑
85	84	ptfinfin 23023	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑑 ∈ PtFin ∧ 𝑥 ∈ ∪ 𝑑) → {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ∈ Fin)
86	85	expcom 415	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 ∈ ∪ 𝑑 → (𝑑 ∈ PtFin → {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ∈ Fin))
87	83, 86	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ PtFin → {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ∈ Fin))
88		simprl 770	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)))
89		elun1 4177	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ 𝑠 → 𝑥 ∈ (𝑠 ∪ 𝑝))
90	89	ad2antll 728	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ (𝑠 ∪ 𝑝))
91	90	ralrimivw 3151	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∀𝑝 ∈ 𝑎 𝑥 ∈ (𝑠 ∪ 𝑝))
92	50	rgenw 3066	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∀𝑝 ∈ 𝑎 (𝑠 ∪ 𝑝) ∈ V
93		eqid 2733	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) = (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))
94		eleq2 2823	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 = (𝑠 ∪ 𝑝) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ (𝑠 ∪ 𝑝)))
95	93, 94	ralrnmptw 7096	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (∀𝑝 ∈ 𝑎 (𝑠 ∪ 𝑝) ∈ V → (∀𝑧 ∈ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧 ↔ ∀𝑝 ∈ 𝑎 𝑥 ∈ (𝑠 ∪ 𝑝)))
96	92, 95	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (∀𝑧 ∈ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧 ↔ ∀𝑝 ∈ 𝑎 𝑥 ∈ (𝑠 ∪ 𝑝))
97	91, 96	sylibr 233	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∀𝑧 ∈ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧)
98	97	adantr 482	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → ∀𝑧 ∈ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧)
99		ssralv 4051	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → (∀𝑧 ∈ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧 → ∀𝑧 ∈ 𝑑 𝑥 ∈ 𝑧))
100	88, 98, 99	sylc 65	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → ∀𝑧 ∈ 𝑑 𝑥 ∈ 𝑧)
101		rabid2 3465	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 = {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ↔ ∀𝑧 ∈ 𝑑 𝑥 ∈ 𝑧)
102	100, 101	sylibr 233	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑑 = {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧})
103	102	eleq1d 2819	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ Fin ↔ {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ∈ Fin))
104	103	biimprd 247	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → ({𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ∈ Fin → 𝑑 ∈ Fin))
105	93	rnmpt 5955	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) = {𝑞 ∣ ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)}
106	88, 105	sseqtrdi 4033	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑑 ⊆ {𝑞 ∣ ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)})
107		ssabral 4060	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 ⊆ {𝑞 ∣ ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)} ↔ ∀𝑞 ∈ 𝑑 ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝))
108	106, 107	sylib 217	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → ∀𝑞 ∈ 𝑑 ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝))
109		uneq2 4158	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = (𝑓‘𝑞) → (𝑠 ∪ 𝑝) = (𝑠 ∪ (𝑓‘𝑞)))
110	109	eqeq2d 2744	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = (𝑓‘𝑞) → (𝑞 = (𝑠 ∪ 𝑝) ↔ 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))
111	110	ac6sfi 9287	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ Fin ∧ ∀𝑞 ∈ 𝑑 ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)) → ∃𝑓(𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))
112	111	expcom 415	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑞 ∈ 𝑑 ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝) → (𝑑 ∈ Fin → ∃𝑓(𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞)))))
113	108, 112	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ Fin → ∃𝑓(𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞)))))
114		frn 6725	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓:𝑑⟶𝑎 → ran 𝑓 ⊆ 𝑎)
115	114	adantr 482	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))) → ran 𝑓 ⊆ 𝑎)
116	115	ad2antll 728	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ran 𝑓 ⊆ 𝑎)
117	32	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑠 ∈ 𝑎)
118	117	snssd 4813	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → {𝑠} ⊆ 𝑎)
119	116, 118	unssd 4187	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → (ran 𝑓 ∪ {𝑠}) ⊆ 𝑎)
120		simprl 770	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑑 ∈ Fin)
121		simprrl 780	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑓:𝑑⟶𝑎)
122	121	ffnd 6719	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑓 Fn 𝑑)
123		dffn4 6812	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 Fn 𝑑 ↔ 𝑓:𝑑–onto→ran 𝑓)
124	122, 123	sylib 217	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑓:𝑑–onto→ran 𝑓)
125		fofi 9338	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑑 ∈ Fin ∧ 𝑓:𝑑–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
126	120, 124, 125	syl2anc 585	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ran 𝑓 ∈ Fin)
127		snfi 9044	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {𝑠} ∈ Fin
128		unfi 9172	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ran 𝑓 ∈ Fin ∧ {𝑠} ∈ Fin) → (ran 𝑓 ∪ {𝑠}) ∈ Fin)
129	126, 127, 128	sylancl 587	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → (ran 𝑓 ∪ {𝑠}) ∈ Fin)
130		elfpw 9354	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ↔ ((ran 𝑓 ∪ {𝑠}) ⊆ 𝑎 ∧ (ran 𝑓 ∪ {𝑠}) ∈ Fin))
131	119, 129, 130	sylanbrc 584	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → (ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin))
132		simplrr 777	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑋 = ∪ 𝑑)
133		uniiun 5062	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ∪ 𝑑 = ∪ 𝑞 ∈ 𝑑 𝑞
134		simprrr 781	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞)))
135		iuneq2 5017	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞)) → ∪ 𝑞 ∈ 𝑑 𝑞 = ∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)))
136	134, 135	syl 17	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ∪ 𝑞 ∈ 𝑑 𝑞 = ∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)))
137	133, 136	eqtrid 2785	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ∪ 𝑑 = ∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)))
138	132, 137	eqtrd 2773	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑋 = ∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)))
139		ssun2 4174	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ {𝑠} ⊆ (ran 𝑓 ∪ {𝑠})
140		vsnid 4666	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝑠 ∈ {𝑠}
141	139, 140	sselii 3980	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑠 ∈ (ran 𝑓 ∪ {𝑠})
142		elssuni 4942	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑠 ∈ (ran 𝑓 ∪ {𝑠}) → 𝑠 ⊆ ∪ (ran 𝑓 ∪ {𝑠}))
143	141, 142	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑠 ⊆ ∪ (ran 𝑓 ∪ {𝑠})
144		fvssunirn 6925	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓‘𝑞) ⊆ ∪ ran 𝑓
145		ssun1 4173	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑠})
146	145	unissi 4918	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ∪ ran 𝑓 ⊆ ∪ (ran 𝑓 ∪ {𝑠})
147	144, 146	sstri 3992	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓‘𝑞) ⊆ ∪ (ran 𝑓 ∪ {𝑠})
148	143, 147	unssi 4186	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran 𝑓 ∪ {𝑠})
149	148	rgenw 3066	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ∀𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran 𝑓 ∪ {𝑠})
150		iunss 5049	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran 𝑓 ∪ {𝑠}) ↔ ∀𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran 𝑓 ∪ {𝑠}))
151	149, 150	mpbir 230	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran 𝑓 ∪ {𝑠})
152	138, 151	eqsstrdi 4037	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑋 ⊆ ∪ (ran 𝑓 ∪ {𝑠}))
153	31	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑎 ⊆ 𝐽)
154	116, 153	sstrd 3993	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ran 𝑓 ⊆ 𝐽)
155	33	ad2antrr 725	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑠 ∈ 𝐽)
156	155	snssd 4813	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → {𝑠} ⊆ 𝐽)
157	154, 156	unssd 4187	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → (ran 𝑓 ∪ {𝑠}) ⊆ 𝐽)
158		uniss 4917	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ran 𝑓 ∪ {𝑠}) ⊆ 𝐽 → ∪ (ran 𝑓 ∪ {𝑠}) ⊆ ∪ 𝐽)
159	158, 2	sseqtrrdi 4034	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ran 𝑓 ∪ {𝑠}) ⊆ 𝐽 → ∪ (ran 𝑓 ∪ {𝑠}) ⊆ 𝑋)
160	157, 159	syl 17	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ∪ (ran 𝑓 ∪ {𝑠}) ⊆ 𝑋)
161	152, 160	eqssd 4000	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → 𝑋 = ∪ (ran 𝑓 ∪ {𝑠}))
162		unieq 4920	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = (ran 𝑓 ∪ {𝑠}) → ∪ 𝑏 = ∪ (ran 𝑓 ∪ {𝑠}))
163	162	rspceeqv 3634	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = ∪ (ran 𝑓 ∪ {𝑠})) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)
164	131, 161, 163	syl2anc 585	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)
165	164	expr 458	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ 𝑑 ∈ Fin) → ((𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))
166	165	exlimdv 1937	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) ∧ 𝑑 ∈ Fin) → (∃𝑓(𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))
167	166	ex 414	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ Fin → (∃𝑓(𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))
168	113, 167	mpdd 43	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ Fin → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))
169	87, 104, 168	3syld 60	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ PtFin → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))
170	169	ex 414	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ((𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑) → (𝑑 ∈ PtFin → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))
171	170	com23 86	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (𝑑 ∈ PtFin → ((𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))
172	171	rexlimdv 3154	. . . . . . . . . . . . 13 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))
173	73, 172	syld 47	. . . . . . . . . . . 12 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))
174	173	rexlimdvaa 3157	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (∃𝑠 ∈ 𝑎 𝑥 ∈ 𝑠 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))
175	30, 174	syld 47	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑥 ∈ 𝑋 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))
176	175	exlimdv 1937	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (∃𝑥 𝑥 ∈ 𝑋 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))
177	25, 176	biimtrid 241	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑋 ≠ ∅ → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))
178	24, 177	pm2.61dne 3029	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))
179	15, 178	syl3an3 1166	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ∈ 𝒫 𝐽) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))
180	179	3exp 1120	. . . . 5 ⊢ (𝐽 ∈ Top → (𝑋 = ∪ 𝑎 → (𝑎 ∈ 𝒫 𝐽 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))))
181	180	com24 95	. . . 4 ⊢ (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → (𝑎 ∈ 𝒫 𝐽 → (𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏))))
182	181	ralrimdv 3153	. . 3 ⊢ (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → ∀𝑎 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))
183	2	iscmp 22892	. . . 4 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))
184	183	baibr 538	. . 3 ⊢ (𝐽 ∈ Top → (∀𝑎 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) ↔ 𝐽 ∈ Comp))
185	182, 184	sylibd 238	. 2 ⊢ (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → 𝐽 ∈ Comp))
186	14, 185	impbid2 225	1 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {cab 2710   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  {crab 3433  Vcvv 3475   ∪ cun 3947   ∩ cin 3948   ⊆ wss 3949  ∅c0 4323  𝒫 cpw 4603  {csn 4629  ∪ cuni 4909  ∪ ciun 4998   ↦ cmpt 5232  ran crn 5678   Fn wfn 6539  ⟶wf 6540  –onto→wfo 6542  ‘cfv 6544  Fincfn 8939  Topctop 22395  Compccmp 22890  PtFincptfin 23007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-om 7856  df-1o 8466  df-er 8703  df-en 8940  df-dom 8941  df-fin 8943  df-top 22396  df-cmp 22891  df-ptfin 23010

This theorem is referenced by: (None)