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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 (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 4435 . . . 4 (𝑐 ∈ 𝒫 𝐽𝑐𝐽)
2 comppfsc.1 . . . . . . 7 𝑋 = 𝐽
32cmpcov 21716 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝑐𝐽𝑋 = 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)
4 elfpw 8627 . . . . . . . 8 (𝑑 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑑𝑐𝑑 ∈ Fin))
5 finptfin 21845 . . . . . . . . . . 11 (𝑑 ∈ Fin → 𝑑 ∈ PtFin)
65anim1i 606 . . . . . . . . . 10 ((𝑑 ∈ Fin ∧ (𝑑𝑐𝑋 = 𝑑)) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
76anassrs 460 . . . . . . . . 9 (((𝑑 ∈ Fin ∧ 𝑑𝑐) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
87ancom1s 641 . . . . . . . 8 (((𝑑𝑐𝑑 ∈ Fin) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
94, 8sylanb 573 . . . . . . 7 ((𝑑 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
109reximi2 3193 . . . . . 6 (∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))
113, 10syl 17 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑐𝐽𝑋 = 𝑐) → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))
12113exp 1100 . . . 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 3133 . 2 (𝐽 ∈ Comp → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)))
15 elpwi 4435 . . . . . . 7 (𝑎 ∈ 𝒫 𝐽𝑎𝐽)
16 0elpw 5114 . . . . . . . . . . 11 ∅ ∈ 𝒫 𝑎
17 0fin 8547 . . . . . . . . . . 11 ∅ ∈ Fin
1816, 17elini 4061 . . . . . . . . . 10 ∅ ∈ (𝒫 𝑎 ∩ Fin)
19 unieq 4725 . . . . . . . . . . . 12 (𝑏 = ∅ → 𝑏 = ∅)
20 uni0 4744 . . . . . . . . . . . 12 ∅ = ∅
2119, 20syl6eq 2832 . . . . . . . . . . 11 (𝑏 = ∅ → 𝑏 = ∅)
2221rspceeqv 3555 . . . . . . . . . 10 ((∅ ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = ∅) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)
2318, 22mpan 678 . . . . . . . . 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 4199 . . . . . . . . 9 (𝑋 ≠ ∅ ↔ ∃𝑥 𝑥𝑋)
26 simp2 1118 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → 𝑋 = 𝑎)
2726eleq2d 2853 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋𝑥 𝑎))
2827biimpd 221 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋𝑥 𝑎))
29 eluni2 4721 . . . . . . . . . . . 12 (𝑥 𝑎 ↔ ∃𝑠𝑎 𝑥𝑠)
3028, 29syl6ib 243 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋 → ∃𝑠𝑎 𝑥𝑠))
31 simpl3 1174 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑎𝐽)
32 simprl 759 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝑎)
3331, 32sseldd 3861 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝐽)
34 elssuni 4746 . . . . . . . . . . . . . . . . . . . . 21 (𝑠𝐽𝑠 𝐽)
3534, 2syl6sseqr 3910 . . . . . . . . . . . . . . . . . . . 20 (𝑠𝐽𝑠𝑋)
3633, 35syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝑋)
3736ralrimivw 3135 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ∀𝑝𝑎 𝑠𝑋)
38 iunss 4840 . . . . . . . . . . . . . . . . . 18 ( 𝑝𝑎 𝑠𝑋 ↔ ∀𝑝𝑎 𝑠𝑋)
3937, 38sylibr 226 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑝𝑎 𝑠𝑋)
40 ssequn1 4047 . . . . . . . . . . . . . . . . 17 ( 𝑝𝑎 𝑠𝑋 ↔ ( 𝑝𝑎 𝑠𝑋) = 𝑋)
4139, 40sylib 210 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ( 𝑝𝑎 𝑠𝑋) = 𝑋)
42 simpl2 1173 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = 𝑎)
43 uniiun 4853 . . . . . . . . . . . . . . . . . 18 𝑎 = 𝑝𝑎 𝑝
4442, 43syl6eq 2832 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = 𝑝𝑎 𝑝)
4544uneq2d 4030 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ( 𝑝𝑎 𝑠𝑋) = ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝))
4641, 45eqtr3d 2818 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝))
47 iunun 4886 . . . . . . . . . . . . . . . 16 𝑝𝑎 (𝑠𝑝) = ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝)
48 vex 3420 . . . . . . . . . . . . . . . . . 18 𝑠 ∈ V
49 vex 3420 . . . . . . . . . . . . . . . . . 18 𝑝 ∈ V
5048, 49unex 7292 . . . . . . . . . . . . . . . . 17 (𝑠𝑝) ∈ V
5150dfiun3 5684 . . . . . . . . . . . . . . . 16 𝑝𝑎 (𝑠𝑝) = ran (𝑝𝑎 ↦ (𝑠𝑝))
5247, 51eqtr3i 2806 . . . . . . . . . . . . . . 15 ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝) = ran (𝑝𝑎 ↦ (𝑠𝑝))
5346, 52syl6eq 2832 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝)))
54 simpll1 1193 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → 𝐽 ∈ Top)
5533adantr 473 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → 𝑠𝐽)
5631sselda 3860 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → 𝑝𝐽)
57 unopn 21230 . . . . . . . . . . . . . . . . . . 19 ((𝐽 ∈ Top ∧ 𝑠𝐽𝑝𝐽) → (𝑠𝑝) ∈ 𝐽)
5854, 55, 56, 57syl3anc 1352 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → (𝑠𝑝) ∈ 𝐽)
5958fmpttd 6708 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (𝑝𝑎 ↦ (𝑠𝑝)):𝑎𝐽)
6059frnd 6356 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽)
61 elpw2g 5107 . . . . . . . . . . . . . . . . . 18 (𝐽 ∈ Top → (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽))
62613ad2ant1 1114 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽))
6362adantr 473 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽))
6460, 63mpbird 249 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽)
65 unieq 4725 . . . . . . . . . . . . . . . . . 18 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → 𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)))
6665eqeq2d 2790 . . . . . . . . . . . . . . . . 17 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → (𝑋 = 𝑐𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝))))
67 sseq2 3885 . . . . . . . . . . . . . . . . . . 19 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → (𝑑𝑐𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝))))
6867anbi1d 621 . . . . . . . . . . . . . . . . . 18 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ((𝑑𝑐𝑋 = 𝑑) ↔ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)))
6968rexbidv 3244 . . . . . . . . . . . . . . . . 17 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → (∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑) ↔ ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)))
7066, 69imbi12d 337 . . . . . . . . . . . . . . . 16 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ((𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) ↔ (𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑))))
7170rspcv 3533 . . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥𝑠)
75 ssel2 3855 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐽𝑠𝑎) → 𝑠𝐽)
76753ad2antl3 1168 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ 𝑠𝑎) → 𝑠𝐽)
7776adantrr 705 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝐽)
78 elunii 4722 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑠𝑠𝐽) → 𝑥 𝐽)
7974, 77, 78syl2anc 576 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥 𝐽)
8079, 2syl6eleqr 2879 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥𝑋)
8180adantr 473 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑥𝑋)
82 simprr 761 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑋 = 𝑑)
8381, 82eleqtrd 2870 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑥 𝑑)
84 eqid 2780 . . . . . . . . . . . . . . . . . . . 20 𝑑 = 𝑑
8584ptfinfin 21846 . . . . . . . . . . . . . . . . . . 19 ((𝑑 ∈ PtFin ∧ 𝑥 𝑑) → {𝑧𝑑𝑥𝑧} ∈ Fin)
8685expcom 406 . . . . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)))
89 elun1 4043 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝑠𝑥 ∈ (𝑠𝑝))
9089ad2antll 717 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥 ∈ (𝑠𝑝))
9190ralrimivw 3135 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ∀𝑝𝑎 𝑥 ∈ (𝑠𝑝))
9250rgenw 3102 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑝𝑎 (𝑠𝑝) ∈ V
93 eqid 2780 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝𝑎 ↦ (𝑠𝑝)) = (𝑝𝑎 ↦ (𝑠𝑝))
94 eleq2 2856 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = (𝑠𝑝) → (𝑥𝑧𝑥 ∈ (𝑠𝑝)))
9593, 94ralrnmpt 6691 . . . . . . . . . . . . . . . . . . . . . . . 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 226 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧)
9897adantr 473 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧)
99 ssralv 3925 . . . . . . . . . . . . . . . . . . . . 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 3322 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = {𝑧𝑑𝑥𝑧} ↔ ∀𝑧𝑑 𝑥𝑧)
102100, 101sylibr 226 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑑 = {𝑧𝑑𝑥𝑧})
103102eleq1d 2852 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ Fin ↔ {𝑧𝑑𝑥𝑧} ∈ Fin))
104103biimprd 240 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ({𝑧𝑑𝑥𝑧} ∈ Fin → 𝑑 ∈ Fin))
10593rnmpt 5675 . . . . . . . . . . . . . . . . . . . . 21 ran (𝑝𝑎 ↦ (𝑠𝑝)) = {𝑞 ∣ ∃𝑝𝑎 𝑞 = (𝑠𝑝)}
10688, 105syl6sseq 3909 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑑 ⊆ {𝑞 ∣ ∃𝑝𝑎 𝑞 = (𝑠𝑝)})
107 ssabral 3934 . . . . . . . . . . . . . . . . . . . 20 (𝑑 ⊆ {𝑞 ∣ ∃𝑝𝑎 𝑞 = (𝑠𝑝)} ↔ ∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝))
108106, 107sylib 210 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝))
109 uneq2 4024 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑓𝑞) → (𝑠𝑝) = (𝑠 ∪ (𝑓𝑞)))
110109eqeq2d 2790 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑓𝑞) → (𝑞 = (𝑠𝑝) ↔ 𝑞 = (𝑠 ∪ (𝑓𝑞))))
111110ac6sfi 8563 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ Fin ∧ ∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝)) → ∃𝑓(𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))
112111expcom 406 . . . . . . . . . . . . . . . . . . 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 6355 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓:𝑑𝑎 → ran 𝑓𝑎)
115114adantr 473 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))) → ran 𝑓𝑎)
116115ad2antll 717 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ran 𝑓𝑎)
11732ad2antrr 714 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑠𝑎)
118117snssd 4621 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → {𝑠} ⊆ 𝑎)
119116, 118unssd 4053 . . . . . . . . . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑑 ∈ Fin)
121 simprrl 769 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑓:𝑑𝑎)
122121ffnd 6350 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6430 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓 Fn 𝑑𝑓:𝑑onto→ran 𝑓)
124122, 123sylib 210 . . . . . . . . . . . . . . . . . . . . . . . . 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 8611 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑑 ∈ Fin ∧ 𝑓:𝑑onto→ran 𝑓) → ran 𝑓 ∈ Fin)
126120, 124, 125syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . . 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 8397 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑠} ∈ Fin
128 unfi 8586 . . . . . . . . . . . . . . . . . . . . . . . 24 ((ran 𝑓 ∈ Fin ∧ {𝑠} ∈ Fin) → (ran 𝑓 ∪ {𝑠}) ∈ Fin)
129126, 127, 128sylancl 578 . . . . . . . . . . . . . . . . . . . . . . 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 8627 . . . . . . . . . . . . . . . . . . . . . . 23 ((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ↔ ((ran 𝑓 ∪ {𝑠}) ⊆ 𝑎 ∧ (ran 𝑓 ∪ {𝑠}) ∈ Fin))
131119, 129, 130sylanbrc 575 . . . . . . . . . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑋 = 𝑑)
133 uniiun 4853 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑑 = 𝑞𝑑 𝑞
134 simprrr 770 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4815 . . . . . . . . . . . . . . . . . . . . . . . . . . 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, 136syl5eq 2828 . . . . . . . . . . . . . . . . . . . . . . . . 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 2816 . . . . . . . . . . . . . . . . . . . . . . . 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 4040 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 {𝑠} ⊆ (ran 𝑓 ∪ {𝑠})
140 vsnid 4479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑠 ∈ {𝑠}
141139, 140sselii 3857 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑠 ∈ (ran 𝑓 ∪ {𝑠})
142 elssuni 4746 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑠 ∈ (ran 𝑓 ∪ {𝑠}) → 𝑠 (ran 𝑓 ∪ {𝑠}))
143141, 142ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑠 (ran 𝑓 ∪ {𝑠})
144 fvssunirn 6533 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓𝑞) ⊆ ran 𝑓
145 ssun1 4039 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑠})
146145unissi 4740 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ran 𝑓 (ran 𝑓 ∪ {𝑠})
147144, 146sstri 3869 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓𝑞) ⊆ (ran 𝑓 ∪ {𝑠})
148143, 147unssi 4052 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠})
149148rgenw 3102 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠})
150 iunss 4840 . . . . . . . . . . . . . . . . . . . . . . . . 25 ( 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠}) ↔ ∀𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠}))
151149, 150mpbir 223 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠})
152138, 151syl6eqss 3913 . . . . . . . . . . . . . . . . . . . . . . 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 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑎𝐽)
154116, 153sstrd 3870 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ran 𝑓𝐽)
15533ad2antrr 714 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑠𝐽)
156155snssd 4621 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → {𝑠} ⊆ 𝐽)
157154, 156unssd 4053 . . . . . . . . . . . . . . . . . . . . . . . 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 4738 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ran 𝑓 ∪ {𝑠}) ⊆ 𝐽 (ran 𝑓 ∪ {𝑠}) ⊆ 𝐽)
159158, 2syl6sseqr 3910 . . . . . . . . . . . . . . . . . . . . . . . 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 3877 . . . . . . . . . . . . . . . . . . . . . 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 4725 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = (ran 𝑓 ∪ {𝑠}) → 𝑏 = (ran 𝑓 ∪ {𝑠}))
163162rspceeqv 3555 . . . . . . . . . . . . . . . . . . . . . 22 (((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = (ran 𝑓 ∪ {𝑠})) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)
164131, 161, 163syl2anc 576 . . . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . . 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 1893 . . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . . . . 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 405 . . . . . . . . . . . . . . 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 3230 . . . . . . . . . . . . 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 3232 . . . . . . . . . . 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 1893 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (∃𝑥 𝑥𝑋 → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
17725, 176syl5bi 234 . . . . . . . 8 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑋 ≠ ∅ → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
17824, 177pm2.61dne 3056 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
17915, 178syl3an3 1146 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎 ∈ 𝒫 𝐽) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
1801793exp 1100 . . . . 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 3140 . . 3 (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∀𝑎 ∈ 𝒫 𝐽(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
1832iscmp 21715 . . . 4 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝒫 𝐽(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
184183baibr 529 . . 3 (𝐽 ∈ Top → (∀𝑎 ∈ 𝒫 𝐽(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏) ↔ 𝐽 ∈ Comp))
185182, 184sylibd 231 . 2 (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → 𝐽 ∈ Comp))
18614, 185impbid2 218 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 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  ontowfo 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)
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