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Theorem comppfsc 22899
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 4572 . . . 4 (𝑐 ∈ 𝒫 𝐽𝑐𝐽)
2 comppfsc.1 . . . . . . 7 𝑋 = 𝐽
32cmpcov 22756 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝑐𝐽𝑋 = 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)
4 elfpw 9305 . . . . . . . 8 (𝑑 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑑𝑐𝑑 ∈ Fin))
5 finptfin 22885 . . . . . . . . . . 11 (𝑑 ∈ Fin → 𝑑 ∈ PtFin)
65anim1i 616 . . . . . . . . . 10 ((𝑑 ∈ Fin ∧ (𝑑𝑐𝑋 = 𝑑)) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
76anassrs 469 . . . . . . . . 9 (((𝑑 ∈ Fin ∧ 𝑑𝑐) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
87ancom1s 652 . . . . . . . 8 (((𝑑𝑐𝑑 ∈ Fin) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
94, 8sylanb 582 . . . . . . 7 ((𝑑 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
109reximi2 3083 . . . . . 6 (∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))
113, 10syl 17 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑐𝐽𝑋 = 𝑐) → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))
12113exp 1120 . . . 4 (𝐽 ∈ Comp → (𝑐𝐽 → (𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))))
131, 12syl5 34 . . 3 (𝐽 ∈ Comp → (𝑐 ∈ 𝒫 𝐽 → (𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))))
1413ralrimiv 3143 . 2 (𝐽 ∈ Comp → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)))
15 elpwi 4572 . . . . . . 7 (𝑎 ∈ 𝒫 𝐽𝑎𝐽)
16 0elpw 5316 . . . . . . . . . . 11 ∅ ∈ 𝒫 𝑎
17 0fin 9122 . . . . . . . . . . 11 ∅ ∈ Fin
1816, 17elini 4158 . . . . . . . . . 10 ∅ ∈ (𝒫 𝑎 ∩ Fin)
19 unieq 4881 . . . . . . . . . . . 12 (𝑏 = ∅ → 𝑏 = ∅)
20 uni0 4901 . . . . . . . . . . . 12 ∅ = ∅
2119, 20eqtrdi 2793 . . . . . . . . . . 11 (𝑏 = ∅ → 𝑏 = ∅)
2221rspceeqv 3600 . . . . . . . . . 10 ((∅ ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = ∅) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)
2318, 22mpan 689 . . . . . . . . 9 (𝑋 = ∅ → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)
2423a1i13 27 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑋 = ∅ → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
25 n0 4311 . . . . . . . . 9 (𝑋 ≠ ∅ ↔ ∃𝑥 𝑥𝑋)
26 simp2 1138 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → 𝑋 = 𝑎)
2726eleq2d 2824 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋𝑥 𝑎))
2827biimpd 228 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋𝑥 𝑎))
29 eluni2 4874 . . . . . . . . . . . 12 (𝑥 𝑎 ↔ ∃𝑠𝑎 𝑥𝑠)
3028, 29syl6ib 251 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋 → ∃𝑠𝑎 𝑥𝑠))
31 simpl3 1194 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑎𝐽)
32 simprl 770 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝑎)
3331, 32sseldd 3950 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝐽)
34 elssuni 4903 . . . . . . . . . . . . . . . . . . . . 21 (𝑠𝐽𝑠 𝐽)
3534, 2sseqtrrdi 4000 . . . . . . . . . . . . . . . . . . . 20 (𝑠𝐽𝑠𝑋)
3633, 35syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝑋)
3736ralrimivw 3148 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ∀𝑝𝑎 𝑠𝑋)
38 iunss 5010 . . . . . . . . . . . . . . . . . 18 ( 𝑝𝑎 𝑠𝑋 ↔ ∀𝑝𝑎 𝑠𝑋)
3937, 38sylibr 233 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑝𝑎 𝑠𝑋)
40 ssequn1 4145 . . . . . . . . . . . . . . . . 17 ( 𝑝𝑎 𝑠𝑋 ↔ ( 𝑝𝑎 𝑠𝑋) = 𝑋)
4139, 40sylib 217 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ( 𝑝𝑎 𝑠𝑋) = 𝑋)
42 simpl2 1193 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = 𝑎)
43 uniiun 5023 . . . . . . . . . . . . . . . . . 18 𝑎 = 𝑝𝑎 𝑝
4442, 43eqtrdi 2793 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = 𝑝𝑎 𝑝)
4544uneq2d 4128 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ( 𝑝𝑎 𝑠𝑋) = ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝))
4641, 45eqtr3d 2779 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝))
47 iunun 5058 . . . . . . . . . . . . . . . 16 𝑝𝑎 (𝑠𝑝) = ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝)
48 vex 3452 . . . . . . . . . . . . . . . . . 18 𝑠 ∈ V
49 vex 3452 . . . . . . . . . . . . . . . . . 18 𝑝 ∈ V
5048, 49unex 7685 . . . . . . . . . . . . . . . . 17 (𝑠𝑝) ∈ V
5150dfiun3 5926 . . . . . . . . . . . . . . . 16 𝑝𝑎 (𝑠𝑝) = ran (𝑝𝑎 ↦ (𝑠𝑝))
5247, 51eqtr3i 2767 . . . . . . . . . . . . . . 15 ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝) = ran (𝑝𝑎 ↦ (𝑠𝑝))
5346, 52eqtrdi 2793 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝)))
54 simpll1 1213 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → 𝐽 ∈ Top)
5533adantr 482 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → 𝑠𝐽)
5631sselda 3949 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → 𝑝𝐽)
57 unopn 22268 . . . . . . . . . . . . . . . . . . 19 ((𝐽 ∈ Top ∧ 𝑠𝐽𝑝𝐽) → (𝑠𝑝) ∈ 𝐽)
5854, 55, 56, 57syl3anc 1372 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → (𝑠𝑝) ∈ 𝐽)
5958fmpttd 7068 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (𝑝𝑎 ↦ (𝑠𝑝)):𝑎𝐽)
6059frnd 6681 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽)
61 elpw2g 5306 . . . . . . . . . . . . . . . . . 18 (𝐽 ∈ Top → (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽))
62613ad2ant1 1134 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽))
6362adantr 482 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽))
6460, 63mpbird 257 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽)
65 unieq 4881 . . . . . . . . . . . . . . . . . 18 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → 𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)))
6665eqeq2d 2748 . . . . . . . . . . . . . . . . 17 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → (𝑋 = 𝑐𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝))))
67 sseq2 3975 . . . . . . . . . . . . . . . . . . 19 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → (𝑑𝑐𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝))))
6867anbi1d 631 . . . . . . . . . . . . . . . . . 18 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ((𝑑𝑐𝑋 = 𝑑) ↔ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)))
6968rexbidv 3176 . . . . . . . . . . . . . . . . 17 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → (∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑) ↔ ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)))
7066, 69imbi12d 345 . . . . . . . . . . . . . . . 16 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ((𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) ↔ (𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑))))
7170rspcv 3580 . . . . . . . . . . . . . . 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))))
7264, 71syl 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))))
7353, 72mpid 44 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)))
74 simprr 772 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥𝑠)
75 ssel2 3944 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐽𝑠𝑎) → 𝑠𝐽)
76753ad2antl3 1188 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ 𝑠𝑎) → 𝑠𝐽)
7776adantrr 716 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝐽)
78 elunii 4875 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑠𝑠𝐽) → 𝑥 𝐽)
7974, 77, 78syl2anc 585 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥 𝐽)
8079, 2eleqtrrdi 2849 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥𝑋)
8180adantr 482 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑥𝑋)
82 simprr 772 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑋 = 𝑑)
8381, 82eleqtrd 2840 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑥 𝑑)
84 eqid 2737 . . . . . . . . . . . . . . . . . . . 20 𝑑 = 𝑑
8584ptfinfin 22886 . . . . . . . . . . . . . . . . . . 19 ((𝑑 ∈ PtFin ∧ 𝑥 𝑑) → {𝑧𝑑𝑥𝑧} ∈ Fin)
8685expcom 415 . . . . . . . . . . . . . . . . . 18 (𝑥 𝑑 → (𝑑 ∈ PtFin → {𝑧𝑑𝑥𝑧} ∈ Fin))
8783, 86syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ PtFin → {𝑧𝑑𝑥𝑧} ∈ Fin))
88 simprl 770 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)))
89 elun1 4141 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝑠𝑥 ∈ (𝑠𝑝))
9089ad2antll 728 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥 ∈ (𝑠𝑝))
9190ralrimivw 3148 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ∀𝑝𝑎 𝑥 ∈ (𝑠𝑝))
9250rgenw 3069 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑝𝑎 (𝑠𝑝) ∈ V
93 eqid 2737 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝𝑎 ↦ (𝑠𝑝)) = (𝑝𝑎 ↦ (𝑠𝑝))
94 eleq2 2827 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = (𝑠𝑝) → (𝑥𝑧𝑥 ∈ (𝑠𝑝)))
9593, 94ralrnmptw 7049 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑝𝑎 (𝑠𝑝) ∈ V → (∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧 ↔ ∀𝑝𝑎 𝑥 ∈ (𝑠𝑝)))
9692, 95ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧 ↔ ∀𝑝𝑎 𝑥 ∈ (𝑠𝑝))
9791, 96sylibr 233 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧)
9897adantr 482 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧)
99 ssralv 4015 . . . . . . . . . . . . . . . . . . . . 21 (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) → (∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧 → ∀𝑧𝑑 𝑥𝑧))
10088, 98, 99sylc 65 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ∀𝑧𝑑 𝑥𝑧)
101 rabid2 3439 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = {𝑧𝑑𝑥𝑧} ↔ ∀𝑧𝑑 𝑥𝑧)
102100, 101sylibr 233 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑑 = {𝑧𝑑𝑥𝑧})
103102eleq1d 2823 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ Fin ↔ {𝑧𝑑𝑥𝑧} ∈ Fin))
104103biimprd 248 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ({𝑧𝑑𝑥𝑧} ∈ Fin → 𝑑 ∈ Fin))
10593rnmpt 5915 . . . . . . . . . . . . . . . . . . . . 21 ran (𝑝𝑎 ↦ (𝑠𝑝)) = {𝑞 ∣ ∃𝑝𝑎 𝑞 = (𝑠𝑝)}
10688, 105sseqtrdi 3999 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑑 ⊆ {𝑞 ∣ ∃𝑝𝑎 𝑞 = (𝑠𝑝)})
107 ssabral 4024 . . . . . . . . . . . . . . . . . . . 20 (𝑑 ⊆ {𝑞 ∣ ∃𝑝𝑎 𝑞 = (𝑠𝑝)} ↔ ∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝))
108106, 107sylib 217 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝))
109 uneq2 4122 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑓𝑞) → (𝑠𝑝) = (𝑠 ∪ (𝑓𝑞)))
110109eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑓𝑞) → (𝑞 = (𝑠𝑝) ↔ 𝑞 = (𝑠 ∪ (𝑓𝑞))))
111110ac6sfi 9238 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ Fin ∧ ∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝)) → ∃𝑓(𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))
112111expcom 415 . . . . . . . . . . . . . . . . . . 19 (∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝) → (𝑑 ∈ Fin → ∃𝑓(𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞)))))
113108, 112syl 17 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ Fin → ∃𝑓(𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞)))))
114 frn 6680 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓:𝑑𝑎 → ran 𝑓𝑎)
115114adantr 482 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))) → ran 𝑓𝑎)
116115ad2antll 728 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ran 𝑓𝑎)
11732ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑠𝑎)
118117snssd 4774 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → {𝑠} ⊆ 𝑎)
119116, 118unssd 4151 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → (ran 𝑓 ∪ {𝑠}) ⊆ 𝑎)
120 simprl 770 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑑 ∈ Fin)
121 simprrl 780 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑓:𝑑𝑎)
122121ffnd 6674 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑓 Fn 𝑑)
123 dffn4 6767 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 Fn 𝑑𝑓:𝑑onto→ran 𝑓)
124122, 123sylib 217 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑓:𝑑onto→ran 𝑓)
125 fofi 9289 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑑 ∈ Fin ∧ 𝑓:𝑑onto→ran 𝑓) → ran 𝑓 ∈ Fin)
126120, 124, 125syl2anc 585 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ran 𝑓 ∈ Fin)
127 snfi 8995 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑠} ∈ Fin
128 unfi 9123 . . . . . . . . . . . . . . . . . . . . . . . 24 ((ran 𝑓 ∈ Fin ∧ {𝑠} ∈ Fin) → (ran 𝑓 ∪ {𝑠}) ∈ Fin)
129126, 127, 128sylancl 587 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → (ran 𝑓 ∪ {𝑠}) ∈ Fin)
130 elfpw 9305 . . . . . . . . . . . . . . . . . . . . . . 23 ((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ↔ ((ran 𝑓 ∪ {𝑠}) ⊆ 𝑎 ∧ (ran 𝑓 ∪ {𝑠}) ∈ Fin))
131119, 129, 130sylanbrc 584 . . . . . . . . . . . . . . . . . . . . . 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))
132 simplrr 777 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑋 = 𝑑)
133 uniiun 5023 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑑 = 𝑞𝑑 𝑞
134 simprrr 781 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞)))
135 iuneq2 4978 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞)) → 𝑞𝑑 𝑞 = 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)))
136134, 135syl 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)))
137133, 136eqtrid 2789 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑑 = 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)))
138132, 137eqtrd 2777 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑋 = 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)))
139 ssun2 4138 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 {𝑠} ⊆ (ran 𝑓 ∪ {𝑠})
140 vsnid 4628 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑠 ∈ {𝑠}
141139, 140sselii 3946 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑠 ∈ (ran 𝑓 ∪ {𝑠})
142 elssuni 4903 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑠 ∈ (ran 𝑓 ∪ {𝑠}) → 𝑠 (ran 𝑓 ∪ {𝑠}))
143141, 142ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑠 (ran 𝑓 ∪ {𝑠})
144 fvssunirn 6880 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓𝑞) ⊆ ran 𝑓
145 ssun1 4137 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑠})
146145unissi 4879 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ran 𝑓 (ran 𝑓 ∪ {𝑠})
147144, 146sstri 3958 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓𝑞) ⊆ (ran 𝑓 ∪ {𝑠})
148143, 147unssi 4150 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠})
149148rgenw 3069 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠})
150 iunss 5010 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠}) ↔ ∀𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠}))
151149, 150mpbir 230 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠})
152138, 151eqsstrdi 4003 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑋 (ran 𝑓 ∪ {𝑠}))
15331ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑎𝐽)
154116, 153sstrd 3959 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ran 𝑓𝐽)
15533ad2antrr 725 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑠𝐽)
156155snssd 4774 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → {𝑠} ⊆ 𝐽)
157154, 156unssd 4151 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → (ran 𝑓 ∪ {𝑠}) ⊆ 𝐽)
158 uniss 4878 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ran 𝑓 ∪ {𝑠}) ⊆ 𝐽 (ran 𝑓 ∪ {𝑠}) ⊆ 𝐽)
159158, 2sseqtrrdi 4000 . . . . . . . . . . . . . . . . . . . . . . . 24 ((ran 𝑓 ∪ {𝑠}) ⊆ 𝐽 (ran 𝑓 ∪ {𝑠}) ⊆ 𝑋)
160157, 159syl 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)
161152, 160eqssd 3966 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑋 = (ran 𝑓 ∪ {𝑠}))
162 unieq 4881 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = (ran 𝑓 ∪ {𝑠}) → 𝑏 = (ran 𝑓 ∪ {𝑠}))
163162rspceeqv 3600 . . . . . . . . . . . . . . . . . . . . . 22 (((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = (ran 𝑓 ∪ {𝑠})) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)
164131, 161, 163syl2anc 585 . . . . . . . . . . . . . . . . . . . . 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)
165164expr 458 . . . . . . . . . . . . . . . . . . . 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))
166165exlimdv 1937 . . . . . . . . . . . . . . . . . . 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))
167166ex 414 . . . . . . . . . . . . . . . . . 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)))
168113, 167mpdd 43 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ Fin → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
16987, 104, 1683syld 60 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ PtFin → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
170169ex 414 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ((𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
171170com23 86 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (𝑑 ∈ PtFin → ((𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
172171rexlimdv 3151 . . . . . . . . . . . . 13 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
17373, 172syld 47 . . . . . . . . . . . 12 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
174173rexlimdvaa 3154 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (∃𝑠𝑎 𝑥𝑠 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
17530, 174syld 47 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
176175exlimdv 1937 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (∃𝑥 𝑥𝑋 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
17725, 176biimtrid 241 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑋 ≠ ∅ → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
17824, 177pm2.61dne 3032 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
17915, 178syl3an3 1166 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎 ∈ 𝒫 𝐽) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
1801793exp 1120 . . . . 5 (𝐽 ∈ Top → (𝑋 = 𝑎 → (𝑎 ∈ 𝒫 𝐽 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))))
181180com24 95 . . . 4 (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → (𝑎 ∈ 𝒫 𝐽 → (𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))))
182181ralrimdv 3150 . . 3 (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∀𝑎 ∈ 𝒫 𝐽(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
1832iscmp 22755 . . . 4 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝒫 𝐽(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
184183baibr 538 . . 3 (𝐽 ∈ Top → (∀𝑎 ∈ 𝒫 𝐽(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) ↔ 𝐽 ∈ Comp))
185182, 184sylibd 238 . 2 (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → 𝐽 ∈ Comp))
18614, 185impbid2 225 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 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2714  wne 2944  wral 3065  wrex 3074  {crab 3410  Vcvv 3448  cun 3913  cin 3914  wss 3915  c0 4287  𝒫 cpw 4565  {csn 4591   cuni 4870   ciun 4959  cmpt 5193  ran crn 5639   Fn wfn 6496  wf 6497  ontowfo 6499  cfv 6501  Fincfn 8890  Topctop 22258  Compccmp 22753  PtFincptfin 22870
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 2708  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-fin 8894  df-top 22259  df-cmp 22754  df-ptfin 22873
This theorem is referenced by: (None)
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