Theorem comppfsc 21859

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 4435	. . . 4 ⊢ (𝑐 ∈ 𝒫 𝐽 → 𝑐 ⊆ 𝐽)
2		comppfsc.1	. . . . . . 7 ⊢ 𝑋 = ∪ 𝐽
3	2	cmpcov 21716	. . . . . 6 ⊢ ((𝐽 ∈ Comp ∧ 𝑐 ⊆ 𝐽 ∧ 𝑋 = ∪ 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = ∪ 𝑑)
4		elfpw 8627	. . . . . . . 8 ⊢ (𝑑 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑑 ⊆ 𝑐 ∧ 𝑑 ∈ Fin))
5		finptfin 21845	. . . . . . . . . . 11 ⊢ (𝑑 ∈ Fin → 𝑑 ∈ PtFin)
6	5	anim1i 606	. . . . . . . . . 10 ⊢ ((𝑑 ∈ Fin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))
7	6	anassrs 460	. . . . . . . . 9 ⊢ (((𝑑 ∈ Fin ∧ 𝑑 ⊆ 𝑐) ∧ 𝑋 = ∪ 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))
8	7	ancom1s 641	. . . . . . . 8 ⊢ (((𝑑 ⊆ 𝑐 ∧ 𝑑 ∈ Fin) ∧ 𝑋 = ∪ 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))
9	4, 8	sylanb 573	. . . . . . 7 ⊢ ((𝑑 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = ∪ 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))
10	9	reximi2 3193	. . . . . 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 1100	. . . 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 3133	. 2 ⊢ (𝐽 ∈ Comp → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)))
15		elpwi 4435	. . . . . . 7 ⊢ (𝑎 ∈ 𝒫 𝐽 → 𝑎 ⊆ 𝐽)
16		0elpw 5114	. . . . . . . . . . 11 ⊢ ∅ ∈ 𝒫 𝑎
17		0fin 8547	. . . . . . . . . . 11 ⊢ ∅ ∈ Fin
18	16, 17	elini 4061	. . . . . . . . . 10 ⊢ ∅ ∈ (𝒫 𝑎 ∩ Fin)
19		unieq 4725	. . . . . . . . . . . 12 ⊢ (𝑏 = ∅ → ∪ 𝑏 = ∪ ∅)
20		uni0 4744	. . . . . . . . . . . 12 ⊢ ∪ ∅ = ∅
21	19, 20	syl6eq 2832	. . . . . . . . . . 11 ⊢ (𝑏 = ∅ → ∪ 𝑏 = ∅)
22	21	rspceeqv 3555	. . . . . . . . . 10 ⊢ ((∅ ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = ∅) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)
23	18, 22	mpan 678	. . . . . . . . 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 4199	. . . . . . . . 9 ⊢ (𝑋 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝑋)
26		simp2 1118	. . . . . . . . . . . . . 14 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → 𝑋 = ∪ 𝑎)
27	26	eleq2d 2853	. . . . . . . . . . . . 13 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝑎))
28	27	biimpd 221	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑥 ∈ 𝑋 → 𝑥 ∈ ∪ 𝑎))
29		eluni2 4721	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ∪ 𝑎 ↔ ∃𝑠 ∈ 𝑎 𝑥 ∈ 𝑠)
30	28, 29	syl6ib 243	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (𝑥 ∈ 𝑋 → ∃𝑠 ∈ 𝑎 𝑥 ∈ 𝑠))
31		simpl3 1174	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑎 ⊆ 𝐽)
32		simprl 759	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ∈ 𝑎)
33	31, 32	sseldd 3861	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ∈ 𝐽)
34		elssuni 4746	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑠 ∈ 𝐽 → 𝑠 ⊆ ∪ 𝐽)
35	34, 2	syl6sseqr 3910	. . . . . . . . . . . . . . . . . . . 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 3135	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∀𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋)
38		iunss 4840	. . . . . . . . . . . . . . . . . 18 ⊢ (∪ 𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋 ↔ ∀𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋)
39	37, 38	sylibr 226	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∪ 𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋)
40		ssequn1 4047	. . . . . . . . . . . . . . . . 17 ⊢ (∪ 𝑝 ∈ 𝑎 𝑠 ⊆ 𝑋 ↔ (∪ 𝑝 ∈ 𝑎 𝑠 ∪ 𝑋) = 𝑋)
41	39, 40	sylib 210	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (∪ 𝑝 ∈ 𝑎 𝑠 ∪ 𝑋) = 𝑋)
42		simpl2 1173	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = ∪ 𝑎)
43		uniiun 4853	. . . . . . . . . . . . . . . . . 18 ⊢ ∪ 𝑎 = ∪ 𝑝 ∈ 𝑎 𝑝
44	42, 43	syl6eq 2832	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = ∪ 𝑝 ∈ 𝑎 𝑝)
45	44	uneq2d 4030	. . . . . . . . . . . . . . . 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 2818	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = (∪ 𝑝 ∈ 𝑎 𝑠 ∪ ∪ 𝑝 ∈ 𝑎 𝑝))
47		iunun 4886	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑝 ∈ 𝑎 (𝑠 ∪ 𝑝) = (∪ 𝑝 ∈ 𝑎 𝑠 ∪ ∪ 𝑝 ∈ 𝑎 𝑝)
48		vex 3420	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑠 ∈ V
49		vex 3420	. . . . . . . . . . . . . . . . . 18 ⊢ 𝑝 ∈ V
50	48, 49	unex 7292	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∪ 𝑝) ∈ V
51	50	dfiun3 5684	. . . . . . . . . . . . . . . 16 ⊢ ∪ 𝑝 ∈ 𝑎 (𝑠 ∪ 𝑝) = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))
52	47, 51	eqtr3i 2806	. . . . . . . . . . . . . . 15 ⊢ (∪ 𝑝 ∈ 𝑎 𝑠 ∪ ∪ 𝑝 ∈ 𝑎 𝑝) = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))
53	46, 52	syl6eq 2832	. . . . . . . . . . . . . 14 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑋 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)))
54		simpll1 1193	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → 𝐽 ∈ Top)
55	33	adantr 473	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → 𝑠 ∈ 𝐽)
56	31	sselda 3860	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → 𝑝 ∈ 𝐽)
57		unopn 21230	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ Top ∧ 𝑠 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) → (𝑠 ∪ 𝑝) ∈ 𝐽)
58	54, 55, 56, 57	syl3anc 1352	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ 𝑝 ∈ 𝑎) → (𝑠 ∪ 𝑝) ∈ 𝐽)
59	58	fmpttd 6708	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)):𝑎⟶𝐽)
60	59	frnd 6356	. . . . . . . . . . . . . . . 16 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽)
61		elpw2g 5107	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐽 ∈ Top → (ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽))
62	61	3ad2ant1 1114	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) → (ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ⊆ 𝐽))
63	62	adantr 473	. . . . . . . . . . . . . . . 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 249	. . . . . . . . . . . . . . 15 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∈ 𝒫 𝐽)
65		unieq 4725	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ∪ 𝑐 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)))
66	65	eqeq2d 2790	. . . . . . . . . . . . . . . . 17 ⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → (𝑋 = ∪ 𝑐 ↔ 𝑋 = ∪ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))))
67		sseq2 3885	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → (𝑑 ⊆ 𝑐 ↔ 𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))))
68	67	anbi1d 621	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑐 = ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) → ((𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑) ↔ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)))
69	68	rexbidv 3244	. . . . . . . . . . . . . . . . 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 337	. . . . . . . . . . . . . . . 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 3533	. . . . . . . . . . . . . . 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 761	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ 𝑠)
75		ssel2 3855	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ⊆ 𝐽 ∧ 𝑠 ∈ 𝑎) → 𝑠 ∈ 𝐽)
76	75	3ad2antl3 1168	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ 𝑠 ∈ 𝑎) → 𝑠 ∈ 𝐽)
77	76	adantrr 705	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑠 ∈ 𝐽)
78		elunii 4722	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑥 ∈ 𝑠 ∧ 𝑠 ∈ 𝐽) → 𝑥 ∈ ∪ 𝐽)
79	74, 77, 78	syl2anc 576	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ ∪ 𝐽)
80	79, 2	syl6eleqr 2879	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ 𝑋)
81	80	adantr 473	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑥 ∈ 𝑋)
82		simprr 761	. . . . . . . . . . . . . . . . . . 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 2870	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) ∧ (𝑑 ⊆ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) ∧ 𝑋 = ∪ 𝑑)) → 𝑥 ∈ ∪ 𝑑)
84		eqid 2780	. . . . . . . . . . . . . . . . . . . 20 ⊢ ∪ 𝑑 = ∪ 𝑑
85	84	ptfinfin 21846	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑑 ∈ PtFin ∧ 𝑥 ∈ ∪ 𝑑) → {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ∈ Fin)
86	85	expcom 406	. . . . . . . . . . . . . . . . . 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 759	. . . . . . . . . . . . . . . . . . . . 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 4043	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥 ∈ 𝑠 → 𝑥 ∈ (𝑠 ∪ 𝑝))
90	89	ad2antll 717	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → 𝑥 ∈ (𝑠 ∪ 𝑝))
91	90	ralrimivw 3135	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∀𝑝 ∈ 𝑎 𝑥 ∈ (𝑠 ∪ 𝑝))
92	50	rgenw 3102	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∀𝑝 ∈ 𝑎 (𝑠 ∪ 𝑝) ∈ V
93		eqid 2780	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) = (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))
94		eleq2 2856	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑧 = (𝑠 ∪ 𝑝) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ (𝑠 ∪ 𝑝)))
95	93, 94	ralrnmpt 6691	. . . . . . . . . . . . . . . . . . . . . . . 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 226	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝐽 ∈ Top ∧ 𝑋 = ∪ 𝑎 ∧ 𝑎 ⊆ 𝐽) ∧ (𝑠 ∈ 𝑎 ∧ 𝑥 ∈ 𝑠)) → ∀𝑧 ∈ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝))𝑥 ∈ 𝑧)
98	97	adantr 473	. . . . . . . . . . . . . . . . . . . . 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 3925	. . . . . . . . . . . . . . . . . . . . 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 3322	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 = {𝑧 ∈ 𝑑 ∣ 𝑥 ∈ 𝑧} ↔ ∀𝑧 ∈ 𝑑 𝑥 ∈ 𝑧)
102	100, 101	sylibr 226	. . . . . . . . . . . . . . . . . . 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 2852	. . . . . . . . . . . . . . . . . 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 240	. . . . . . . . . . . . . . . . 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 5675	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ran (𝑝 ∈ 𝑎 ↦ (𝑠 ∪ 𝑝)) = {𝑞 ∣ ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)}
106	88, 105	syl6sseq 3909	. . . . . . . . . . . . . . . . . . . 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 3934	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑑 ⊆ {𝑞 ∣ ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)} ↔ ∀𝑞 ∈ 𝑑 ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝))
108	106, 107	sylib 210	. . . . . . . . . . . . . . . . . . 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 4024	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝 = (𝑓‘𝑞) → (𝑠 ∪ 𝑝) = (𝑠 ∪ (𝑓‘𝑞)))
110	109	eqeq2d 2790	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑝 = (𝑓‘𝑞) → (𝑞 = (𝑠 ∪ 𝑝) ↔ 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))
111	110	ac6sfi 8563	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑑 ∈ Fin ∧ ∀𝑞 ∈ 𝑑 ∃𝑝 ∈ 𝑎 𝑞 = (𝑠 ∪ 𝑝)) → ∃𝑓(𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))))
112	111	expcom 406	. . . . . . . . . . . . . . . . . . 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 6355	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓:𝑑⟶𝑎 → ran 𝑓 ⊆ 𝑎)
115	114	adantr 473	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑓:𝑑⟶𝑎 ∧ ∀𝑞 ∈ 𝑑 𝑞 = (𝑠 ∪ (𝑓‘𝑞))) → ran 𝑓 ⊆ 𝑎)
116	115	ad2antll 717	. . . . . . . . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . . . . . . . . . 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 4621	. . . . . . . . . . . . . . . . . . . . . . . 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 4053	. . . . . . . . . . . . . . . . . . . . . . 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 759	. . . . . . . . . . . . . . . . . . . . . . . . 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 769	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 6350	. . . . . . . . . . . . . . . . . . . . . . . . . 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 6430	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑓 Fn 𝑑 ↔ 𝑓:𝑑–onto→ran 𝑓)
124	122, 123	sylib 210	. . . . . . . . . . . . . . . . . . . . . . . . 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 8611	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑑 ∈ Fin ∧ 𝑓:𝑑–onto→ran 𝑓) → ran 𝑓 ∈ Fin)
126	120, 124, 125	syl2anc 576	. . . . . . . . . . . . . . . . . . . . . . . 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 8397	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ {𝑠} ∈ Fin
128		unfi 8586	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((ran 𝑓 ∈ Fin ∧ {𝑠} ∈ Fin) → (ran 𝑓 ∪ {𝑠}) ∈ Fin)
129	126, 127, 128	sylancl 578	. . . . . . . . . . . . . . . . . . . . . . 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 8627	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ↔ ((ran 𝑓 ∪ {𝑠}) ⊆ 𝑎 ∧ (ran 𝑓 ∪ {𝑠}) ∈ Fin))
131	119, 129, 130	sylanbrc 575	. . . . . . . . . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . . . . . . . . . . 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 4853	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ∪ 𝑑 = ∪ 𝑞 ∈ 𝑑 𝑞
134		simprrr 770	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 4815	. . . . . . . . . . . . . . . . . . . . . . . . . . 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	syl5eq 2828	. . . . . . . . . . . . . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . . . . . . . . . . . . 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 4040	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ {𝑠} ⊆ (ran 𝑓 ∪ {𝑠})
140		vsnid 4479	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ 𝑠 ∈ {𝑠}
141	139, 140	sselii 3857	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ 𝑠 ∈ (ran 𝑓 ∪ {𝑠})
142		elssuni 4746	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑠 ∈ (ran 𝑓 ∪ {𝑠}) → 𝑠 ⊆ ∪ (ran 𝑓 ∪ {𝑠}))
143	141, 142	ax-mp 5	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ 𝑠 ⊆ ∪ (ran 𝑓 ∪ {𝑠})
144		fvssunirn 6533	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (𝑓‘𝑞) ⊆ ∪ ran 𝑓
145		ssun1 4039	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑠})
146	145	unissi 4740	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ∪ ran 𝑓 ⊆ ∪ (ran 𝑓 ∪ {𝑠})
147	144, 146	sstri 3869	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑓‘𝑞) ⊆ ∪ (ran 𝑓 ∪ {𝑠})
148	143, 147	unssi 4052	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran 𝑓 ∪ {𝑠})
149	148	rgenw 3102	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ∀𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran 𝑓 ∪ {𝑠})
150		iunss 4840	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran 𝑓 ∪ {𝑠}) ↔ ∀𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran 𝑓 ∪ {𝑠}))
151	149, 150	mpbir 223	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ∪ 𝑞 ∈ 𝑑 (𝑠 ∪ (𝑓‘𝑞)) ⊆ ∪ (ran 𝑓 ∪ {𝑠})
152	138, 151	syl6eqss 3913	. . . . . . . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . . . . . . . . . . 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 3870	. . . . . . . . . . . . . . . . . . . . . . . . 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 714	. . . . . . . . . . . . . . . . . . . . . . . . . 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 4621	. . . . . . . . . . . . . . . . . . . . . . . . 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 4053	. . . . . . . . . . . . . . . . . . . . . . . 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 4738	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((ran 𝑓 ∪ {𝑠}) ⊆ 𝐽 → ∪ (ran 𝑓 ∪ {𝑠}) ⊆ ∪ 𝐽)
159	158, 2	syl6sseqr 3910	. . . . . . . . . . . . . . . . . . . . . . . 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 3877	. . . . . . . . . . . . . . . . . . . . . 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 4725	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑏 = (ran 𝑓 ∪ {𝑠}) → ∪ 𝑏 = ∪ (ran 𝑓 ∪ {𝑠}))
163	162	rspceeqv 3555	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = ∪ (ran 𝑓 ∪ {𝑠})) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)
164	131, 161, 163	syl2anc 576	. . . . . . . . . . . . . . . . . . . . 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 449	. . . . . . . . . . . . . . . . . . . 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 1893	. . . . . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . . . . 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 405	. . . . . . . . . . . . . . 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 3230	. . . . . . . . . . . . 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 3232	. . . . . . . . . . 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 1893	. . . . . . . . 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	syl5bi 234	. . . . . . . 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 3056	. . . . . . 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 1146	. . . . . 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 1100	. . . . 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 3140	. . 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 21715	. . . 4 ⊢ (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏)))
184	183	baibr 529	. . 3 ⊢ (𝐽 ∈ Top → (∀𝑎 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = ∪ 𝑏) ↔ 𝐽 ∈ Comp))
185	182, 184	sylibd 231	. 2 ⊢ (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑)) → 𝐽 ∈ Comp))
186	14, 185	impbid2 218	1 ⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = ∪ 𝑐 → ∃𝑑 ∈ PtFin (𝑑 ⊆ 𝑐 ∧ 𝑋 = ∪ 𝑑))))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1069   = wceq 1508  ∃wex 1743   ∈ wcel 2051  {cab 2760   ≠ wne 2969  ∀wral 3090  ∃wrex 3091  {crab 3094  Vcvv 3417   ∪ cun 3829   ∩ cin 3830   ⊆ wss 3831  ∅c0 4181  𝒫 cpw 4425  {csn 4444  ∪ cuni 4717  ∪ ciun 4797   ↦ cmpt 5013  ran crn 5412   Fn wfn 6188  ⟶wf 6189  –onto→wfo 6191  ‘cfv 6193  Fincfn 8312  Topctop 21220  Compccmp 21713  PtFincptfin 21830

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2752  ax-sep 5064  ax-nul 5071  ax-pow 5123  ax-pr 5190  ax-un 7285

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2551  df-eu 2589  df-clab 2761  df-cleq 2773  df-clel 2848  df-nfc 2920  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3419  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4182  df-if 4354  df-pw 4427  df-sn 4445  df-pr 4447  df-tp 4449  df-op 4451  df-uni 4718  df-int 4755  df-iun 4799  df-br 4935  df-opab 4997  df-mpt 5014  df-tr 5036  df-id 5316  df-eprel 5321  df-po 5330  df-so 5331  df-fr 5370  df-we 5372  df-xp 5417  df-rel 5418  df-cnv 5419  df-co 5420  df-dm 5421  df-rn 5422  df-res 5423  df-ima 5424  df-pred 5991  df-ord 6037  df-on 6038  df-lim 6039  df-suc 6040  df-iota 6157  df-fun 6195  df-fn 6196  df-f 6197  df-f1 6198  df-fo 6199  df-f1o 6200  df-fv 6201  df-ov 6985  df-oprab 6986  df-mpo 6987  df-om 7403  df-wrecs 7756  df-recs 7818  df-rdg 7856  df-1o 7911  df-oadd 7915  df-er 8095  df-en 8313  df-dom 8314  df-fin 8316  df-top 21221  df-cmp 21714  df-ptfin 21833

This theorem is referenced by: (None)