Theorem comppfsc 23468

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 4582	. . . 4 ⊢ (𝑐 ∈ 𝒫 𝐽 → 𝑐 ⊆ 𝐽)
2		comppfsc.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	cmpcov 23325	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)
4		elfpw 9364	. . . . . . . 8 ⊢ (𝑑 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑑 ⊆ 𝑐 ∧ 𝑑 ∈ Fin))
5		finptfin 23454	. . . . . . . . . . 11 ⊢ (𝑑 ∈ Fin → 𝑑 ∈ PtFin)
6	5	anim1i 615	. . . . . . . . . 10 ⊢ ((𝑑 ∈ Fin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))
7	6	anassrs 467	. . . . . . . . 9 ⊢ (((𝑑 ∈ Fin ∧ 𝑑 ⊆ 𝑐) ∧ 𝑋 = ∪ 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))
8	7	ancom1s 653	. . . . . . . 8 ⊢ (((𝑑 ⊆ 𝑐 ∧ 𝑑 ∈ Fin) ∧ 𝑋 = ∪ 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))
9	4, 8	sylanb 581	. . . . . . 7 ⊢ ((𝑑 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = ∪ 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))
10	9	reximi2 3069	. . . . . 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 1119	. . . 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 3131	. 2 ⊢ (𝐽 ∈ Comp → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))
15		elpwi 4582	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝐽 → 𝑎 ⊆ 𝐽)
16		0elpw 5326	. . . . . . . . . . 11 ⊢ ∅ ∈ 𝒫 𝑎
17		0fi 9054	. . . . . . . . . . 11 ⊢ ∅ ∈ Fin
18	16, 17	elini 4174	. . . . . . . . . 10 ⊢ ∅ ∈ (𝒫 𝑎 ∩ Fin)
19		unieq 4894	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → ∪ 𝑏 = ∪ ∅)
20		uni0 4911	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
21	19, 20	eqtrdi 2786	. . . . . . . . . . 11 ⊢ (𝑏 = ∅ → ∪ 𝑏 = ∅)
22	21	rspceeqv 3624	. . . . . . . . . 10 ⊢ ((∅ ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = ∅) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)
23	18, 22	mpan 690	. . . . . . . . 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 4328	. . . . . . . . 9 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑋)
26		simp2 1137	. . . . . . . . . . . . . 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 229	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑥 ∈ 𝑋 → 𝑥 ∈ ∪ 𝑎))
29		eluni2 4887	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑎 ↔ ∃𝑠 ∈ 𝑎 𝑥 ∈ 𝑠)
30	28, 29	imbitrdi 251	. . . . . . . . . . 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 3959	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ∈ 𝐽)
34		elssuni 4913	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ 𝐽 → 𝑠 ⊆ ∪ 𝐽)
35	34, 2	sseqtrrdi 4000	. . . . . . . . . . . . . . . . . . . 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 3136	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∀𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋)
38		iunss 5021	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋 ↔ ∀𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋)
39	37, 38	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∪ 𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋)
40		ssequn1 4161	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋 ↔ (∪ 𝑝 ∈ 𝑎 𝑠 ∪ 𝑋) = 𝑋)
41	39, 40	sylib 218	. . . . . . . . . . . . . . . 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 5034	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑎 = ∪ 𝑝 ∈ 𝑎 𝑝
44	42, 43	eqtrdi 2786	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = ∪ 𝑝 ∈ 𝑎 𝑝)
45	44	uneq2d 4143	. . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = (∪ 𝑝 ∈ 𝑎 𝑠 ∪ ∪ 𝑝 ∈ 𝑎 𝑝))
47		iunun 5069	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑝 ∈ 𝑎 (𝑠 ∪ 𝑝) = (∪ 𝑝 ∈ 𝑎 𝑠 ∪ ∪ 𝑝 ∈ 𝑎 𝑝)
48		vex 3463	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑠 ∈ V
49		vex 3463	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑝 ∈ V
50	48, 49	unex 7736	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∪ 𝑝) ∈ V
51	50	dfiun3 5949	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑝 ∈ 𝑎 (𝑠 ∪ 𝑝) = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))
52	47, 51	eqtr3i 2760	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑝 ∈ 𝑎 𝑠 ∪ ∪ 𝑝 ∈ 𝑎 𝑝) = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))
53	46, 52	eqtrdi 2786	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → 𝑠 ∈ 𝐽)
56	31	sselda 3958	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → 𝑝 ∈ 𝐽)
57		unopn 22839	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) → (𝑠 ∪ 𝑝) ∈ 𝐽)
58	54, 55, 56, 57	syl3anc 1373	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → (𝑠 ∪ 𝑝) ∈ 𝐽)
59	58	fmpttd 7104	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)):𝑎⟶𝐽)
60	59	frnd 6713	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽)
61		elpw2g 5303	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 ∈ Top → (ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽))
62	61	3ad2ant1 1133	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽))
63	62	adantr 480	. . . . . . . . . . . . . . . 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 4894	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ∪ 𝑐 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)))
66	65	eqeq2d 2746	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → (𝑋 = ∪ 𝑐 ↔ 𝑋 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))))
67		sseq2 3985	. . . . . . . . . . . . . . . . . . 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 3164	. . . . . . . . . . . . . . . . 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 344	. . . . . . . . . . . . . . . 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 3597	. . . . . . . . . . . . . . 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 3953	. . . . . . . . . . . . . . . . . . . . . . . 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 717	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ∈ 𝐽)
78		elunii 4888	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 ∈ 𝐽) → 𝑥 ∈ ∪ 𝐽)
79	74, 77, 78	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 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 480	. . . . . . . . . . . . . . . . . . 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 2735	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑑 = ∪ 𝑑
85	84	ptfinfin 23455	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑑 ∈ PtFin ∧ 𝑥 ∈ ∪ 𝑑) → {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ∈ Fin)
86	85	expcom 413	. . . . . . . . . . . . . . . . . 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 4157	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ 𝑠 → 𝑥 ∈ (𝑠 ∪ 𝑝))
90	89	ad2antll 729	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ (𝑠 ∪ 𝑝))
91	90	ralrimivw 3136	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∀𝑝 ∈ 𝑎 𝑥 ∈ (𝑠 ∪ 𝑝))
92	50	rgenw 3055	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∀𝑝 ∈ 𝑎 (𝑠 ∪ 𝑝) ∈ V
93		eqid 2735	. . . . . . . . . . . . . . . . . . . . . . . . 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 7083	. . . . . . . . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∀𝑧 ∈ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧)
98	97	adantr 480	. . . . . . . . . . . . . . . . . . . . 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 4027	. . . . . . . . . . . . . . . . . . . . 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 3449	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 = {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ↔ ∀𝑧 ∈ 𝑑 𝑥 ∈ 𝑧)
102	100, 101	sylibr 234	. . . . . . . . . . . . . . . . . . 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 248	. . . . . . . . . . . . . . . . 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 5937	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) = {𝑞 ∣ ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)}
106	88, 105	sseqtrdi 3999	. . . . . . . . . . . . . . . . . . . 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 4040	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 ⊆ {𝑞 ∣ ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)} ↔ ∀𝑞 ∈ 𝑑 ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝))
108	106, 107	sylib 218	. . . . . . . . . . . . . . . . . . 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 4137	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = (𝑓‘𝑞) → (𝑠 ∪ 𝑝) = (𝑠 ∪ (𝑓‘𝑞)))
110	109	eqeq2d 2746	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = (𝑓‘𝑞) → (𝑞 = (𝑠 ∪ 𝑝) ↔ 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))
111	110	ac6sfi 9290	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ Fin ∧ ∀𝑞 ∈ 𝑑 ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)) → ∃𝑓(𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))
112	111	expcom 413	. . . . . . . . . . . . . . . . . . 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 6712	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓:𝑑⟶𝑎 → ran 𝑓 ⊆ 𝑎)
115	114	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))) → ran 𝑓 ⊆ 𝑎)
116	115	ad2antll 729	. . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . . . 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 4785	. . . . . . . . . . . . . . . . . . . . . . . 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 4167	. . . . . . . . . . . . . . . . . . . . . . 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 6706	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6795	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 Fn 𝑑 ↔ 𝑓:𝑑–onto→ran 𝑓)
124	122, 123	sylib 218	. . . . . . . . . . . . . . . . . . . . . . . . 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 9321	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑑 ∈ Fin ∧ 𝑓:𝑑–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
126	120, 124, 125	syl2anc 584	. . . . . . . . . . . . . . . . . . . . . . . 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 9055	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {𝑠} ∈ Fin
128		unfi 9183	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ran 𝑓 ∈ Fin ∧ {𝑠} ∈ Fin) → (ran 𝑓 ∪ {𝑠}) ∈ Fin)
129	126, 127, 128	sylancl 586	. . . . . . . . . . . . . . . . . . . . . . 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 9364	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ↔ ((ran 𝑓 ∪ {𝑠}) ⊆ 𝑎 ∧ (ran 𝑓 ∪ {𝑠}) ∈ Fin))
131	119, 129, 130	sylanbrc 583	. . . . . . . . . . . . . . . . . . . . . 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 5034	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4987	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . . . . . . . . . . . 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 2770	. . . . . . . . . . . . . . . . . . . . . . . 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 4154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ {𝑠} ⊆ (ran 𝑓 ∪ {𝑠})
140		vsnid 4639	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝑠 ∈ {𝑠}
141	139, 140	sselii 3955	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑠 ∈ (ran 𝑓 ∪ {𝑠})
142		elssuni 4913	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑠 ∈ (ran 𝑓 ∪ {𝑠}) → 𝑠 ⊆ ∪ (ran 𝑓 ∪ {𝑠}))
143	141, 142	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑠 ⊆ ∪ (ran 𝑓 ∪ {𝑠})
144		fvssunirn 6908	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓‘𝑞) ⊆ ∪ ran 𝑓
145		ssun1 4153	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑠})
146	145	unissi 4892	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ∪ ran 𝑓 ⊆ ∪ (ran 𝑓 ∪ {𝑠})
147	144, 146	sstri 3968	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓‘𝑞) ⊆ ∪ (ran 𝑓 ∪ {𝑠})
148	143, 147	unssi 4166	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran 𝑓 ∪ {𝑠})
149	148	rgenw 3055	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ∀𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran 𝑓 ∪ {𝑠})
150		iunss 5021	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran 𝑓 ∪ {𝑠}) ↔ ∀𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran 𝑓 ∪ {𝑠}))
151	149, 150	mpbir 231	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran 𝑓 ∪ {𝑠})
152	138, 151	eqsstrdi 4003	. . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3969	. . . . . . . . . . . . . . . . . . . . . . . . 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 726	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4785	. . . . . . . . . . . . . . . . . . . . . . . . 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 4167	. . . . . . . . . . . . . . . . . . . . . . . 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 4891	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ran 𝑓 ∪ {𝑠}) ⊆ 𝐽 → ∪ (ran 𝑓 ∪ {𝑠}) ⊆ ∪ 𝐽)
159	158, 2	sseqtrrdi 4000	. . . . . . . . . . . . . . . . . . . . . . . 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 3976	. . . . . . . . . . . . . . . . . . . . . 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 4894	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = (ran 𝑓 ∪ {𝑠}) → ∪ 𝑏 = ∪ (ran 𝑓 ∪ {𝑠}))
163	162	rspceeqv 3624	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = ∪ (ran 𝑓 ∪ {𝑠})) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)
164	131, 161, 163	syl2anc 584	. . . . . . . . . . . . . . . . . . . . 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 456	. . . . . . . . . . . . . . . . . . . 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 1933	. . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . 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 3139	. . . . . . . . . . . . 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 3142	. . . . . . . . . . 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 1933	. . . . . . . . 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 242	. . . . . . . 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 3018	. . . . . . 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 1165	. . . . . 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 1119	. . . . 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 3138	. . 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 23324	. . . 4 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))
184	183	baibr 536	. . 3 ⊢ (𝐽 ∈ Top → (∀𝑎 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) ↔ 𝐽 ∈ Comp))
185	182, 184	sylibd 239	. 2 ⊢ (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → 𝐽 ∈ Comp))
186	14, 185	impbid2 226	1 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2713   ≠ wne 2932  ∀wral 3051  ∃wrex 3060  {crab 3415  Vcvv 3459   ∪ cun 3924   ∩ cin 3925   ⊆ wss 3926  ∅c0 4308  𝒫 cpw 4575  {csn 4601  ∪ cuni 4883  ∪ ciun 4967   ↦ cmpt 5201  ran crn 5655   Fn wfn 6525  ⟶wf 6526  –onto→wfo 6528  ‘cfv 6530  Fincfn 8957  Topctop 22829  Compccmp 23322  PtFincptfin 23439

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-f1 6535  df-fo 6536  df-f1o 6537  df-fv 6538  df-om 7860  df-1o 8478  df-en 8958  df-dom 8959  df-fin 8961  df-top 22830  df-cmp 23323  df-ptfin 23442

This theorem is referenced by: (None)