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Theorem comppfsc 22883
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 4567 . . . 4 (𝑐 ∈ 𝒫 𝐽𝑐𝐽)
2 comppfsc.1 . . . . . . 7 𝑋 = 𝐽
32cmpcov 22740 . . . . . 6 ((𝐽 ∈ Comp ∧ 𝑐𝐽𝑋 = 𝑐) → ∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑)
4 elfpw 9298 . . . . . . . 8 (𝑑 ∈ (𝒫 𝑐 ∩ Fin) ↔ (𝑑𝑐𝑑 ∈ Fin))
5 finptfin 22869 . . . . . . . . . . 11 (𝑑 ∈ Fin → 𝑑 ∈ PtFin)
65anim1i 615 . . . . . . . . . 10 ((𝑑 ∈ Fin ∧ (𝑑𝑐𝑋 = 𝑑)) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
76anassrs 468 . . . . . . . . 9 (((𝑑 ∈ Fin ∧ 𝑑𝑐) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
87ancom1s 651 . . . . . . . 8 (((𝑑𝑐𝑑 ∈ Fin) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
94, 8sylanb 581 . . . . . . 7 ((𝑑 ∈ (𝒫 𝑐 ∩ Fin) ∧ 𝑋 = 𝑑) → (𝑑 ∈ PtFin ∧ (𝑑𝑐𝑋 = 𝑑)))
109reximi2 3082 . . . . . 6 (∃𝑑 ∈ (𝒫 𝑐 ∩ Fin)𝑋 = 𝑑 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))
113, 10syl 17 . . . . 5 ((𝐽 ∈ Comp ∧ 𝑐𝐽𝑋 = 𝑐) → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑))
12113exp 1119 . . . 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 3142 . 2 (𝐽 ∈ Comp → ∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)))
15 elpwi 4567 . . . . . . 7 (𝑎 ∈ 𝒫 𝐽𝑎𝐽)
16 0elpw 5311 . . . . . . . . . . 11 ∅ ∈ 𝒫 𝑎
17 0fin 9115 . . . . . . . . . . 11 ∅ ∈ Fin
1816, 17elini 4153 . . . . . . . . . 10 ∅ ∈ (𝒫 𝑎 ∩ Fin)
19 unieq 4876 . . . . . . . . . . . 12 (𝑏 = ∅ → 𝑏 = ∅)
20 uni0 4896 . . . . . . . . . . . 12 ∅ = ∅
2119, 20eqtrdi 2792 . . . . . . . . . . 11 (𝑏 = ∅ → 𝑏 = ∅)
2221rspceeqv 3595 . . . . . . . . . 10 ((∅ ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = ∅) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)
2318, 22mpan 688 . . . . . . . . 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 4306 . . . . . . . . 9 (𝑋 ≠ ∅ ↔ ∃𝑥 𝑥𝑋)
26 simp2 1137 . . . . . . . . . . . . . 14 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → 𝑋 = 𝑎)
2726eleq2d 2823 . . . . . . . . . . . . 13 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋𝑥 𝑎))
2827biimpd 228 . . . . . . . . . . . 12 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋𝑥 𝑎))
29 eluni2 4869 . . . . . . . . . . . 12 (𝑥 𝑎 ↔ ∃𝑠𝑎 𝑥𝑠)
3028, 29syl6ib 250 . . . . . . . . . . 11 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (𝑥𝑋 → ∃𝑠𝑎 𝑥𝑠))
31 simpl3 1193 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑎𝐽)
32 simprl 769 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝑎)
3331, 32sseldd 3945 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝐽)
34 elssuni 4898 . . . . . . . . . . . . . . . . . . . . 21 (𝑠𝐽𝑠 𝐽)
3534, 2sseqtrrdi 3995 . . . . . . . . . . . . . . . . . . . 20 (𝑠𝐽𝑠𝑋)
3633, 35syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝑋)
3736ralrimivw 3147 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ∀𝑝𝑎 𝑠𝑋)
38 iunss 5005 . . . . . . . . . . . . . . . . . 18 ( 𝑝𝑎 𝑠𝑋 ↔ ∀𝑝𝑎 𝑠𝑋)
3937, 38sylibr 233 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑝𝑎 𝑠𝑋)
40 ssequn1 4140 . . . . . . . . . . . . . . . . 17 ( 𝑝𝑎 𝑠𝑋 ↔ ( 𝑝𝑎 𝑠𝑋) = 𝑋)
4139, 40sylib 217 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ( 𝑝𝑎 𝑠𝑋) = 𝑋)
42 simpl2 1192 . . . . . . . . . . . . . . . . . 18 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = 𝑎)
43 uniiun 5018 . . . . . . . . . . . . . . . . . 18 𝑎 = 𝑝𝑎 𝑝
4442, 43eqtrdi 2792 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = 𝑝𝑎 𝑝)
4544uneq2d 4123 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ( 𝑝𝑎 𝑠𝑋) = ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝))
4641, 45eqtr3d 2778 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝))
47 iunun 5053 . . . . . . . . . . . . . . . 16 𝑝𝑎 (𝑠𝑝) = ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝)
48 vex 3449 . . . . . . . . . . . . . . . . . 18 𝑠 ∈ V
49 vex 3449 . . . . . . . . . . . . . . . . . 18 𝑝 ∈ V
5048, 49unex 7680 . . . . . . . . . . . . . . . . 17 (𝑠𝑝) ∈ V
5150dfiun3 5921 . . . . . . . . . . . . . . . 16 𝑝𝑎 (𝑠𝑝) = ran (𝑝𝑎 ↦ (𝑠𝑝))
5247, 51eqtr3i 2766 . . . . . . . . . . . . . . 15 ( 𝑝𝑎 𝑠 𝑝𝑎 𝑝) = ran (𝑝𝑎 ↦ (𝑠𝑝))
5346, 52eqtrdi 2792 . . . . . . . . . . . . . 14 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝)))
54 simpll1 1212 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → 𝐽 ∈ Top)
5533adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → 𝑠𝐽)
5631sselda 3944 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → 𝑝𝐽)
57 unopn 22252 . . . . . . . . . . . . . . . . . . 19 ((𝐽 ∈ Top ∧ 𝑠𝐽𝑝𝐽) → (𝑠𝑝) ∈ 𝐽)
5854, 55, 56, 57syl3anc 1371 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ 𝑝𝑎) → (𝑠𝑝) ∈ 𝐽)
5958fmpttd 7063 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (𝑝𝑎 ↦ (𝑠𝑝)):𝑎𝐽)
6059frnd 6676 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽)
61 elpw2g 5301 . . . . . . . . . . . . . . . . . 18 (𝐽 ∈ Top → (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽))
62613ad2ant1 1133 . . . . . . . . . . . . . . . . 17 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽))
6362adantr 481 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → (ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽 ↔ ran (𝑝𝑎 ↦ (𝑠𝑝)) ⊆ 𝐽))
6460, 63mpbird 256 . . . . . . . . . . . . . . 15 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ran (𝑝𝑎 ↦ (𝑠𝑝)) ∈ 𝒫 𝐽)
65 unieq 4876 . . . . . . . . . . . . . . . . . 18 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → 𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)))
6665eqeq2d 2747 . . . . . . . . . . . . . . . . 17 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → (𝑋 = 𝑐𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝))))
67 sseq2 3970 . . . . . . . . . . . . . . . . . . 19 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → (𝑑𝑐𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝))))
6867anbi1d 630 . . . . . . . . . . . . . . . . . 18 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ((𝑑𝑐𝑋 = 𝑑) ↔ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)))
6968rexbidv 3175 . . . . . . . . . . . . . . . . 17 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → (∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑) ↔ ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)))
7066, 69imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑐 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ((𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) ↔ (𝑋 = ran (𝑝𝑎 ↦ (𝑠𝑝)) → ∃𝑑 ∈ PtFin (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑))))
7170rspcv 3577 . . . . . . . . . . . . . . 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 771 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥𝑠)
75 ssel2 3939 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝐽𝑠𝑎) → 𝑠𝐽)
76753ad2antl3 1187 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ 𝑠𝑎) → 𝑠𝐽)
7776adantrr 715 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑠𝐽)
78 elunii 4870 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑠𝑠𝐽) → 𝑥 𝐽)
7974, 77, 78syl2anc 584 . . . . . . . . . . . . . . . . . . . . 21 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥 𝐽)
8079, 2eleqtrrdi 2849 . . . . . . . . . . . . . . . . . . . 20 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥𝑋)
8180adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑥𝑋)
82 simprr 771 . . . . . . . . . . . . . . . . . . 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 2736 . . . . . . . . . . . . . . . . . . . 20 𝑑 = 𝑑
8584ptfinfin 22870 . . . . . . . . . . . . . . . . . . 19 ((𝑑 ∈ PtFin ∧ 𝑥 𝑑) → {𝑧𝑑𝑥𝑧} ∈ Fin)
8685expcom 414 . . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)))
89 elun1 4136 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥𝑠𝑥 ∈ (𝑠𝑝))
9089ad2antll 727 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → 𝑥 ∈ (𝑠𝑝))
9190ralrimivw 3147 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) → ∀𝑝𝑎 𝑥 ∈ (𝑠𝑝))
9250rgenw 3068 . . . . . . . . . . . . . . . . . . . . . . . 24 𝑝𝑎 (𝑠𝑝) ∈ V
93 eqid 2736 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝𝑎 ↦ (𝑠𝑝)) = (𝑝𝑎 ↦ (𝑠𝑝))
94 eleq2 2826 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 = (𝑠𝑝) → (𝑥𝑧𝑥 ∈ (𝑠𝑝)))
9593, 94ralrnmptw 7044 . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ∀𝑧 ∈ ran (𝑝𝑎 ↦ (𝑠𝑝))𝑥𝑧)
99 ssralv 4010 . . . . . . . . . . . . . . . . . . . . 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 3436 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = {𝑧𝑑𝑥𝑧} ↔ ∀𝑧𝑑 𝑥𝑧)
102100, 101sylibr 233 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑑 = {𝑧𝑑𝑥𝑧})
103102eleq1d 2822 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → (𝑑 ∈ Fin ↔ {𝑧𝑑𝑥𝑧} ∈ Fin))
104103biimprd 247 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → ({𝑧𝑑𝑥𝑧} ∈ Fin → 𝑑 ∈ Fin))
10593rnmpt 5910 . . . . . . . . . . . . . . . . . . . . 21 ran (𝑝𝑎 ↦ (𝑠𝑝)) = {𝑞 ∣ ∃𝑝𝑎 𝑞 = (𝑠𝑝)}
10688, 105sseqtrdi 3994 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) → 𝑑 ⊆ {𝑞 ∣ ∃𝑝𝑎 𝑞 = (𝑠𝑝)})
107 ssabral 4019 . . . . . . . . . . . . . . . . . . . 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 4117 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = (𝑓𝑞) → (𝑠𝑝) = (𝑠 ∪ (𝑓𝑞)))
110109eqeq2d 2747 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 = (𝑓𝑞) → (𝑞 = (𝑠𝑝) ↔ 𝑞 = (𝑠 ∪ (𝑓𝑞))))
111110ac6sfi 9231 . . . . . . . . . . . . . . . . . . . 20 ((𝑑 ∈ Fin ∧ ∀𝑞𝑑𝑝𝑎 𝑞 = (𝑠𝑝)) → ∃𝑓(𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))
112111expcom 414 . . . . . . . . . . . . . . . . . . 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 6675 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑓:𝑑𝑎 → ran 𝑓𝑎)
115114adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))) → ran 𝑓𝑎)
116115ad2antll 727 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ran 𝑓𝑎)
11732ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑠𝑎)
118117snssd 4769 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → {𝑠} ⊆ 𝑎)
119116, 118unssd 4146 . . . . . . . . . . . . . . . . . . . . . . 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 769 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑑 ∈ Fin)
121 simprrl 779 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑓:𝑑𝑎)
122121ffnd 6669 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6762 . . . . . . . . . . . . . . . . . . . . . . . . . 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 9282 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑑 ∈ Fin ∧ 𝑓:𝑑onto→ran 𝑓) → ran 𝑓 ∈ Fin)
126120, 124, 125syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . 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 8988 . . . . . . . . . . . . . . . . . . . . . . . 24 {𝑠} ∈ Fin
128 unfi 9116 . . . . . . . . . . . . . . . . . . . . . . . 24 ((ran 𝑓 ∈ Fin ∧ {𝑠} ∈ Fin) → (ran 𝑓 ∪ {𝑠}) ∈ Fin)
129126, 127, 128sylancl 586 . . . . . . . . . . . . . . . . . . . . . . 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 9298 . . . . . . . . . . . . . . . . . . . . . . 23 ((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ↔ ((ran 𝑓 ∪ {𝑠}) ⊆ 𝑎 ∧ (ran 𝑓 ∪ {𝑠}) ∈ Fin))
131119, 129, 130sylanbrc 583 . . . . . . . . . . . . . . . . . . . . . 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 776 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑋 = 𝑑)
133 uniiun 5018 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑑 = 𝑞𝑑 𝑞
134 simprrr 780 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4973 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2788 . . . . . . . . . . . . . . . . . . . . . . . . 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 2776 . . . . . . . . . . . . . . . . . . . . . . . 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 4133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 {𝑠} ⊆ (ran 𝑓 ∪ {𝑠})
140 vsnid 4623 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑠 ∈ {𝑠}
141139, 140sselii 3941 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 𝑠 ∈ (ran 𝑓 ∪ {𝑠})
142 elssuni 4898 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑠 ∈ (ran 𝑓 ∪ {𝑠}) → 𝑠 (ran 𝑓 ∪ {𝑠}))
143141, 142ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑠 (ran 𝑓 ∪ {𝑠})
144 fvssunirn 6875 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑓𝑞) ⊆ ran 𝑓
145 ssun1 4132 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ran 𝑓 ⊆ (ran 𝑓 ∪ {𝑠})
146145unissi 4874 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ran 𝑓 (ran 𝑓 ∪ {𝑠})
147144, 146sstri 3953 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑓𝑞) ⊆ (ran 𝑓 ∪ {𝑠})
148143, 147unssi 4145 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠})
149148rgenw 3068 . . . . . . . . . . . . . . . . . . . . . . . . 25 𝑞𝑑 (𝑠 ∪ (𝑓𝑞)) ⊆ (ran 𝑓 ∪ {𝑠})
150 iunss 5005 . . . . . . . . . . . . . . . . . . . . . . . . 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 3998 . . . . . . . . . . . . . . . . . . . . . . 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 724 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑎𝐽)
154116, 153sstrd 3954 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → ran 𝑓𝐽)
15533ad2antrr 724 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → 𝑠𝐽)
156155snssd 4769 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) ∧ (𝑠𝑎𝑥𝑠)) ∧ (𝑑 ⊆ ran (𝑝𝑎 ↦ (𝑠𝑝)) ∧ 𝑋 = 𝑑)) ∧ (𝑑 ∈ Fin ∧ (𝑓:𝑑𝑎 ∧ ∀𝑞𝑑 𝑞 = (𝑠 ∪ (𝑓𝑞))))) → {𝑠} ⊆ 𝐽)
157154, 156unssd 4146 . . . . . . . . . . . . . . . . . . . . . . . 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 4873 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((ran 𝑓 ∪ {𝑠}) ⊆ 𝐽 (ran 𝑓 ∪ {𝑠}) ⊆ 𝐽)
159158, 2sseqtrrdi 3995 . . . . . . . . . . . . . . . . . . . . . . . 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 3961 . . . . . . . . . . . . . . . . . . . . . 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 4876 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = (ran 𝑓 ∪ {𝑠}) → 𝑏 = (ran 𝑓 ∪ {𝑠}))
163162rspceeqv 3595 . . . . . . . . . . . . . . . . . . . . . 22 (((ran 𝑓 ∪ {𝑠}) ∈ (𝒫 𝑎 ∩ Fin) ∧ 𝑋 = (ran 𝑓 ∪ {𝑠})) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)
164131, 161, 163syl2anc 584 . . . . . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . . . . . 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 1936 . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . 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 3150 . . . . . . . . . . . . 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 3153 . . . . . . . . . . 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 1936 . . . . . . . . 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 3031 . . . . . . 7 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎𝐽) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
17915, 178syl3an3 1165 . . . . . 6 ((𝐽 ∈ Top ∧ 𝑋 = 𝑎𝑎 ∈ 𝒫 𝐽) → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏))
1801793exp 1119 . . . . 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 3149 . . 3 (𝐽 ∈ Top → (∀𝑐 ∈ 𝒫 𝐽(𝑋 = 𝑐 → ∃𝑑 ∈ PtFin (𝑑𝑐𝑋 = 𝑑)) → ∀𝑎 ∈ 𝒫 𝐽(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
1832iscmp 22739 . . . 4 (𝐽 ∈ Comp ↔ (𝐽 ∈ Top ∧ ∀𝑎 ∈ 𝒫 𝐽(𝑋 = 𝑎 → ∃𝑏 ∈ (𝒫 𝑎 ∩ Fin)𝑋 = 𝑏)))
184183baibr 537 . . 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 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2713  wne 2943  wral 3064  wrex 3073  {crab 3407  Vcvv 3445  cun 3908  cin 3909  wss 3910  c0 4282  𝒫 cpw 4560  {csn 4586   cuni 4865   ciun 4954  cmpt 5188  ran crn 5634   Fn wfn 6491  wf 6492  ontowfo 6494  cfv 6496  Fincfn 8883  Topctop 22242  Compccmp 22737  PtFincptfin 22854
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-om 7803  df-1o 8412  df-er 8648  df-en 8884  df-dom 8885  df-fin 8887  df-top 22243  df-cmp 22738  df-ptfin 22857
This theorem is referenced by: (None)
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