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Theorem comppfsc 21145
 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 (𝑑𝑐𝑋 = 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.
StepHypRef Expression
1 elpwi 4117 . . . 4 (𝑐 ∈ 𝒫 𝐽𝑐𝐽)
2 comppfsc.1 . . . . . . 7 𝑋 = 𝐽
32cmpcov 21002 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝑐𝐽𝑋 = 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)
4 elfpw 8151 . . . . . . . 8 (𝑑 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑑𝑐𝑑 ∈ Fin))
5 finptfin 21131 . . . . . . . . . . 11 (𝑑 ∈ Fin → 𝑑 ∈ PtFin)
65anim1i 590 . . . . . . . . . 10 ((𝑑 ∈ Fin ∧ (𝑑𝑐𝑋 = 𝑑)) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
76anassrs 678 . . . . . . . . 9 (((𝑑 ∈ Fin ∧ 𝑑𝑐) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
87ancom1s 843 . . . . . . . 8 (((𝑑𝑐𝑑 ∈ Fin) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
94, 8sylanb 488 . . . . . . 7 ((𝑑 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
109reximi2 2993 . . . . . 6 (∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))
113, 10syl 17 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑐𝐽𝑋 = 𝑐) → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))
12113exp 1256 . . . 4 (𝐽 ∈ Comp → (𝑐𝐽 → (𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))))
131, 12syl5 33 . . 3 (𝐽 ∈ Comp → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))))
1413ralrimiv 2948 . 2 (𝐽 ∈ Comp → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)))
15 elpwi 4117 . . . . . . 7 (𝑎 ∈ 𝒫 𝐽𝑎𝐽)
16 0elpw 4760 . . . . . . . . . . . 12 ∅ ∈ 𝒫 𝑎
17 0fin 8073 . . . . . . . . . . . 12 ∅ ∈ Fin
18 elin 3758 . . . . . . . . . . . 12 (∅ ∈ (𝒫 𝑎 ∩ Fin) ↔ (∅ ∈ 𝒫 𝑎 ∧ ∅ ∈ Fin))
1916, 17, 18mpbir2an 957 . . . . . . . . . . 11 ∅ ∈ (𝒫 𝑎 ∩ Fin)
20 unieq 4380 . . . . . . . . . . . . . 14 (𝑏 = ∅ → 𝑏 = ∅)
21 uni0 4401 . . . . . . . . . . . . . 14 ∅ = ∅
2220, 21syl6eq 2660 . . . . . . . . . . . . 13 (𝑏 = ∅ → 𝑏 = ∅)
2322eqeq2d 2620 . . . . . . . . . . . 12 (𝑏 = ∅ → (𝑋 = 𝑏𝑋 = ∅))
2423rspcev 3282 . . . . . . . . . . 11 ((∅ ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = ∅) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)
2519, 24mpan 702 . . . . . . . . . 10 (𝑋 = ∅ → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)
2625a1d 25 . . . . . . . . 9 (𝑋 = ∅ → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
2726a1i 11 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑋 = ∅ → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
28 n0 3890 . . . . . . . . 9 (𝑋 ≠ ∅ ↔ ∃𝑥 𝑥𝑋)
29 simp2 1055 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → 𝑋 = 𝑎)
3029eleq2d 2673 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋𝑥 𝑎))
3130biimpd 218 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋𝑥 𝑎))
32 eluni2 4376 . . . . . . . . . . . 12 (𝑥 𝑎 ↔ ∃𝑠𝑎 𝑥𝑠)
3331, 32syl6ib 240 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋 → ∃𝑠𝑎 𝑥𝑠))
34 simpl3 1059 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑎𝐽)
35 simprl 790 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝑎)
3634, 35sseldd 3569 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝐽)
37 elssuni 4403 . . . . . . . . . . . . . . . . . . . . 21 (𝑠𝐽𝑠 𝐽)
3837, 2syl6sseqr 3615 . . . . . . . . . . . . . . . . . . . 20 (𝑠𝐽𝑠𝑋)
3936, 38syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝑋)
4039ralrimivw 2950 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ∀𝑝𝑎 𝑠𝑋)
41 iunss 4497 . . . . . . . . . . . . . . . . . 18 ( 𝑝𝑎 𝑠𝑋 ↔ ∀𝑝𝑎 𝑠𝑋)
4240, 41sylibr 223 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑝𝑎 𝑠𝑋)
43 ssequn1 3745 . . . . . . . . . . . . . . . . 17 ( 𝑝𝑎 𝑠𝑋 ↔ ( 𝑝𝑎 𝑠𝑋) = 𝑋)
4442, 43sylib 207 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ( 𝑝𝑎 𝑠𝑋) = 𝑋)
45 simpl2 1058 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = 𝑎)
46 uniiun 4509 . . . . . . . . . . . . . . . . . 18 𝑎 = 𝑝𝑎 𝑝
4745, 46syl6eq 2660 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = 𝑝𝑎 𝑝)
4847uneq2d 3729 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ( 𝑝𝑎 𝑠𝑋) = ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝))
4944, 48eqtr3d 2646 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝))
50 iunun 4540 . . . . . . . . . . . . . . . 16 𝑝𝑎 (𝑠𝑝) = ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝)
51 vex 3176 . . . . . . . . . . . . . . . . . 18 𝑠 ∈ V
52 vex 3176 . . . . . . . . . . . . . . . . . 18 𝑝 ∈ V
5351, 52unex 6854 . . . . . . . . . . . . . . . . 17 (𝑠𝑝) ∈ V
5453dfiun3 5301 . . . . . . . . . . . . . . . 16 𝑝𝑎 (𝑠𝑝) = ran (𝑝𝑎 ↦ (𝑠𝑝))
5550, 54eqtr3i 2634 . . . . . . . . . . . . . . 15 ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝) = ran (𝑝𝑎 ↦ (𝑠𝑝))
5649, 55syl6eq 2660 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝)))
57 simpll1 1093 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → 𝐽 ∈ Top)
5836adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → 𝑠𝐽)
5934sselda 3568 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → 𝑝𝐽)
60 unopn 20533 . . . . . . . . . . . . . . . . . . 19 ((𝐽 ∈ Top ∧ 𝑠𝐽𝑝𝐽) → (𝑠𝑝) ∈ 𝐽)
6157, 58, 59, 60syl3anc 1318 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → (𝑠𝑝) ∈ 𝐽)
62 eqid 2610 . . . . . . . . . . . . . . . . . 18 (𝑝𝑎 ↦ (𝑠𝑝)) = (𝑝𝑎 ↦ (𝑠𝑝))
6361, 62fmptd 6292 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (𝑝𝑎 ↦ (𝑠𝑝)):𝑎𝐽)
64 frn 5966 . . . . . . . . . . . . . . . . 17 ((𝑝𝑎 ↦ (𝑠𝑝)):𝑎𝐽 → ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽)
6563, 64syl 17 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽)
66 elpw2g 4754 . . . . . . . . . . . . . . . . . 18 (𝐽 ∈ Top → (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽))
67663ad2ant1 1075 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽))
6867adantr 480 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽))
6965, 68mpbird 246 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽)
70 unieq 4380 . . . . . . . . . . . . . . . . . 18 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → 𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)))
7170eqeq2d 2620 . . . . . . . . . . . . . . . . 17 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → (𝑋 = 𝑐𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝))))
72 sseq2 3590 . . . . . . . . . . . . . . . . . . 19 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → (𝑑𝑐𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝))))
7372anbi1d 737 . . . . . . . . . . . . . . . . . 18 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ((𝑑𝑐𝑋 = 𝑑) ↔ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)))
7473rexbidv 3034 . . . . . . . . . . . . . . . . 17 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → (∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑) ↔ ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)))
7571, 74imbi12d 333 . . . . . . . . . . . . . . . 16 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ((𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) ↔ (𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑))))
7675rspcv 3278 . . . . . . . . . . . . . . 15 (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → (𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑))))
7769, 76syl 17 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → (𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑))))
7856, 77mpid 43 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)))
79 simprr 792 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥𝑠)
80 ssel2 3563 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐽𝑠𝑎) → 𝑠𝐽)
81803ad2antl3 1218 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ 𝑠𝑎) → 𝑠𝐽)
8281adantrr 749 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝐽)
83 elunii 4377 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑠𝑠𝐽) → 𝑥 𝐽)
8479, 82, 83syl2anc 691 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥 𝐽)
8584, 2syl6eleqr 2699 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥𝑋)
8685adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑥𝑋)
87 simprr 792 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑋 = 𝑑)
8886, 87eleqtrd 2690 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑥 𝑑)
89 eqid 2610 . . . . . . . . . . . . . . . . . . . 20 𝑑 = 𝑑
9089ptfinfin 21132 . . . . . . . . . . . . . . . . . . 19 ((𝑑 ∈ PtFin ∧ 𝑥 𝑑) → {𝑧𝑑𝑥𝑧} ∈ Fin)
9190expcom 450 . . . . . . . . . . . . . . . . . 18 (𝑥 𝑑 → (𝑑 ∈ PtFin → {𝑧𝑑𝑥𝑧} ∈ Fin))
9288, 91syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ PtFin → {𝑧𝑑𝑥𝑧} ∈ Fin))
93 simprl 790 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)))
94 elun1 3742 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝑠𝑥 ∈ (𝑠𝑝))
9594ad2antll 761 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥 ∈ (𝑠𝑝))
9695ralrimivw 2950 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ∀𝑝𝑎 𝑥 ∈ (𝑠𝑝))
9753rgenw 2908 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑝𝑎 (𝑠𝑝) ∈ V
98 eleq2 2677 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = (𝑠𝑝) → (𝑥𝑧𝑥 ∈ (𝑠𝑝)))
9962, 98ralrnmpt 6276 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑝𝑎 (𝑠𝑝) ∈ V → (∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧 ↔ ∀𝑝𝑎 𝑥 ∈ (𝑠𝑝)))
10097, 99ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧 ↔ ∀𝑝𝑎 𝑥 ∈ (𝑠𝑝))
10196, 100sylibr 223 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧)
102101adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧)
103 ssralv 3629 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) → (∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧 → ∀𝑧𝑑 𝑥𝑧))
10493, 102, 103sylc 63 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ∀𝑧𝑑 𝑥𝑧)
105 rabid2 3096 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = {𝑧𝑑𝑥𝑧} ↔ ∀𝑧𝑑 𝑥𝑧)
106104, 105sylibr 223 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑑 = {𝑧𝑑𝑥𝑧})
107106eleq1d 2672 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ Fin ↔ {𝑧𝑑𝑥𝑧} ∈ Fin))
108107biimprd 237 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ({𝑧𝑑𝑥𝑧} ∈ Fin → 𝑑 ∈ Fin))
10962rnmpt 5292 . . . . . . . . . . . . . . . . . . . . 21 ran (𝑝𝑎 ↦ (𝑠𝑝)) = {𝑞 ∣ ∃𝑝𝑎 𝑞 = (𝑠𝑝)}
11093, 109syl6sseq 3614 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑑 ⊆ {𝑞 ∣ ∃𝑝𝑎 𝑞 = (𝑠𝑝)})
111 ssabral 3636 . . . . . . . . . . . . . . . . . . . 20 (𝑑 ⊆ {𝑞 ∣ ∃𝑝𝑎 𝑞 = (𝑠𝑝)} ↔ ∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝))
112110, 111sylib 207 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝))
113 uneq2 3723 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑓𝑞) → (𝑠𝑝) = (𝑠 ∪ (𝑓𝑞)))
114113eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑓𝑞) → (𝑞 = (𝑠𝑝) ↔ 𝑞 = (𝑠 ∪ (𝑓𝑞))))
115114ac6sfi 8089 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ Fin ∧ ∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝)) → ∃𝑓(𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))
116115expcom 450 . . . . . . . . . . . . . . . . . . 19 (∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝) → (𝑑 ∈ Fin → ∃𝑓(𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞)))))
117112, 116syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ Fin → ∃𝑓(𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞)))))
118 frn 5966 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓:𝑑𝑎 → ran 𝑓𝑎)
119118adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))) → ran 𝑓𝑎)
120119ad2antll 761 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ran 𝑓𝑎)
12135ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑠𝑎)
122121snssd 4281 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → {𝑠} ⊆ 𝑎)
123120, 122unssd 3751 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → (ran 𝑓 ∪ {𝑠}) ⊆ 𝑎)
124 simprl 790 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑑 ∈ Fin)
125 simprrl 800 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑓:𝑑𝑎)
126 ffn 5958 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓:𝑑𝑎𝑓 Fn 𝑑)
127125, 126syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑓 Fn 𝑑)
128 dffn4 6034 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 Fn 𝑑𝑓:𝑑onto→ran 𝑓)
129127, 128sylib 207 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑓:𝑑onto→ran 𝑓)
130 fofi 8135 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑑 ∈ Fin ∧ 𝑓:𝑑onto→ran 𝑓) → ran 𝑓 ∈ Fin)
131124, 129, 130syl2anc 691 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ran 𝑓 ∈ Fin)
132 snfi 7923 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑠} ∈ Fin
133 unfi 8112 . . . . . . . . . . . . . . . . . . . . . . . 24 ((ran 𝑓 ∈ Fin ∧ {𝑠} ∈ Fin) → (ran 𝑓 ∪ {𝑠}) ∈ Fin)
134131, 132, 133sylancl 693 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → (ran 𝑓 ∪ {𝑠}) ∈ Fin)
135 elfpw 8151 . . . . . . . . . . . . . . . . . . . . . . 23 ((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ↔ ((ran 𝑓 ∪ {𝑠}) ⊆ 𝑎 ∧ (ran 𝑓 ∪ {𝑠}) ∈ Fin))
136123, 134, 135sylanbrc 695 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → (ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin))
137 simplrr 797 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑋 = 𝑑)
138 uniiun 4509 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑑 = 𝑞𝑑 𝑞
139 simprrr 801 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞)))
140 iuneq2 4473 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞)) → 𝑞𝑑 𝑞 = 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)))
141139, 140syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑞𝑑 𝑞 = 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)))
142138, 141syl5eq 2656 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑑 = 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)))
143137, 142eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑋 = 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)))
144 ssun2 3739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 {𝑠} ⊆ (ran 𝑓 ∪ {𝑠})
145 vsnid 4156 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑠 ∈ {𝑠}
146144, 145sselii 3565 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑠 ∈ (ran 𝑓 ∪ {𝑠})
147 elssuni 4403 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑠 ∈ (ran 𝑓 ∪ {𝑠}) → 𝑠 (ran 𝑓 ∪ {𝑠}))
148146, 147ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑠 (ran 𝑓 ∪ {𝑠})
149 fvssunirn 6127 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓𝑞) ⊆ ran 𝑓
150 ssun1 3738 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑠})
151150unissi 4397 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ran 𝑓 (ran 𝑓 ∪ {𝑠})
152149, 151sstri 3577 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓𝑞) ⊆ (ran 𝑓 ∪ {𝑠})
153148, 152unssi 3750 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠})
154153rgenw 2908 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠})
155 iunss 4497 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠}) ↔ ∀𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠}))
156154, 155mpbir 220 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠})
157143, 156syl6eqss 3618 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑋 (ran 𝑓 ∪ {𝑠}))
15834ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑎𝐽)
159120, 158sstrd 3578 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ran 𝑓𝐽)
16036ad2antrr 758 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑠𝐽)
161160snssd 4281 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → {𝑠} ⊆ 𝐽)
162159, 161unssd 3751 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → (ran 𝑓 ∪ {𝑠}) ⊆ 𝐽)
163 uniss 4394 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ran 𝑓 ∪ {𝑠}) ⊆ 𝐽 (ran 𝑓 ∪ {𝑠}) ⊆ 𝐽)
164163, 2syl6sseqr 3615 . . . . . . . . . . . . . . . . . . . . . . . 24 ((ran 𝑓 ∪ {𝑠}) ⊆ 𝐽 (ran 𝑓 ∪ {𝑠}) ⊆ 𝑋)
165162, 164syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → (ran 𝑓 ∪ {𝑠}) ⊆ 𝑋)
166157, 165eqssd 3585 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑋 = (ran 𝑓 ∪ {𝑠}))
167 unieq 4380 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = (ran 𝑓 ∪ {𝑠}) → 𝑏 = (ran 𝑓 ∪ {𝑠}))
168167eqeq2d 2620 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = (ran 𝑓 ∪ {𝑠}) → (𝑋 = 𝑏𝑋 = (ran 𝑓 ∪ {𝑠})))
169168rspcev 3282 . . . . . . . . . . . . . . . . . . . . . 22 (((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = (ran 𝑓 ∪ {𝑠})) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)
170136, 166, 169syl2anc 691 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)
171170expr 641 . . . . . . . . . . . . . . . . . . . 20 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ 𝑑 ∈ Fin) → ((𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
172171exlimdv 1848 . . . . . . . . . . . . . . . . . . 19 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ 𝑑 ∈ Fin) → (∃𝑓(𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
173172ex 449 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ Fin → (∃𝑓(𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
174117, 173mpdd 42 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ Fin → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
17592, 108, 1743syld 58 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ PtFin → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
176175ex 449 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ((𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
177176com23 84 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (𝑑 ∈ PtFin → ((𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
178177rexlimdv 3012 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
17978, 178syld 46 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
180179rexlimdvaa 3014 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (∃𝑠𝑎 𝑥𝑠 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
18133, 180syld 46 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
182181exlimdv 1848 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (∃𝑥 𝑥𝑋 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
18328, 182syl5bi 231 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑋 ≠ ∅ → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
18427, 183pm2.61dne 2868 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
18515, 184syl3an3 1353 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎 ∈ 𝒫 𝐽) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
1861853exp 1256 . . . . 5 (𝐽 ∈ Top → (𝑋 = 𝑎 → (𝑎 ∈ 𝒫 𝐽 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))))
187186com24 93 . . . 4 (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → (𝑎 ∈ 𝒫 𝐽 → (𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))))
188187ralrimdv 2951 . . 3 (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∀𝑎 ∈ 𝒫 𝐽(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
1892iscmp 21001 . . . 4 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝒫 𝐽(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
190189baibr 943 . . 3 (𝐽 ∈ Top → (∀𝑎 ∈ 𝒫 𝐽(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) ↔ 𝐽 ∈ Comp))
191188, 190sylibd 228 . 2 (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → 𝐽 ∈ Comp))
19214, 191impbid2 215 1 (𝐽 ∈ Top → (𝐽 ∈ Comp ↔ ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  𝒫 cpw 4108  {csn 4125  ∪ cuni 4372  ∪ ciun 4455   ↦ cmpt 4643  ran crn 5039   Fn wfn 5799  ⟶wf 5800  –onto→wfo 5802  ‘cfv 5804  Fincfn 7841  Topctop 20517  Compccmp 20999  PtFincptfin 21116 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-en 7842  df-dom 7843  df-fin 7845  df-top 20521  df-cmp 21000  df-ptfin 21119 This theorem is referenced by: (None)
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