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Theorem cncmp 21928
Description: Compactness is respected by a continuous onto map. (Contributed by Jeff Hankins, 12-Jul-2009.) (Proof shortened by Mario Carneiro, 22-Aug-2015.)
Hypothesis
Ref Expression
cncmp.2 𝑌 = 𝐾
Assertion
Ref Expression
cncmp ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp)

Proof of Theorem cncmp
Dummy variables 𝑐 𝑑 𝑠 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cntop2 21777 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
213ad2ant3 1127 . 2 ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3 elpwi 4547 . . . 4 (𝑢 ∈ 𝒫 𝐾𝑢𝐾)
4 simpl1 1183 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐽 ∈ Comp)
5 simpl3 1185 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐹 ∈ (𝐽 Cn 𝐾))
6 simprl 767 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝑢𝐾)
76sselda 3964 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝑦𝐾)
8 cnima 21801 . . . . . . . . . 10 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
95, 7, 8syl2an2r 681 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → (𝐹𝑦) ∈ 𝐽)
109fmpttd 6871 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝑦𝑢 ↦ (𝐹𝑦)):𝑢𝐽)
1110frnd 6514 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ran (𝑦𝑢 ↦ (𝐹𝑦)) ⊆ 𝐽)
12 simprr 769 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝑌 = 𝑢)
1312imaeq2d 5922 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝐹𝑌) = (𝐹 𝑢))
14 eqid 2818 . . . . . . . . . . 11 𝐽 = 𝐽
15 cncmp.2 . . . . . . . . . . 11 𝑌 = 𝐾
1614, 15cnf 21782 . . . . . . . . . 10 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽𝑌)
175, 16syl 17 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐹: 𝐽𝑌)
18 fimacnv 6831 . . . . . . . . 9 (𝐹: 𝐽𝑌 → (𝐹𝑌) = 𝐽)
1917, 18syl 17 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝐹𝑌) = 𝐽)
209ralrimiva 3179 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∀𝑦𝑢 (𝐹𝑦) ∈ 𝐽)
21 dfiun2g 4946 . . . . . . . . . 10 (∀𝑦𝑢 (𝐹𝑦) ∈ 𝐽 𝑦𝑢 (𝐹𝑦) = {𝑥 ∣ ∃𝑦𝑢 𝑥 = (𝐹𝑦)})
2220, 21syl 17 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝑦𝑢 (𝐹𝑦) = {𝑥 ∣ ∃𝑦𝑢 𝑥 = (𝐹𝑦)})
23 imauni 6996 . . . . . . . . 9 (𝐹 𝑢) = 𝑦𝑢 (𝐹𝑦)
24 eqid 2818 . . . . . . . . . . 11 (𝑦𝑢 ↦ (𝐹𝑦)) = (𝑦𝑢 ↦ (𝐹𝑦))
2524rnmpt 5820 . . . . . . . . . 10 ran (𝑦𝑢 ↦ (𝐹𝑦)) = {𝑥 ∣ ∃𝑦𝑢 𝑥 = (𝐹𝑦)}
2625unieqi 4839 . . . . . . . . 9 ran (𝑦𝑢 ↦ (𝐹𝑦)) = {𝑥 ∣ ∃𝑦𝑢 𝑥 = (𝐹𝑦)}
2722, 23, 263eqtr4g 2878 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝐹 𝑢) = ran (𝑦𝑢 ↦ (𝐹𝑦)))
2813, 19, 273eqtr3d 2861 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐽 = ran (𝑦𝑢 ↦ (𝐹𝑦)))
2914cmpcov 21925 . . . . . . 7 ((𝐽 ∈ Comp ∧ ran (𝑦𝑢 ↦ (𝐹𝑦)) ⊆ 𝐽 𝐽 = ran (𝑦𝑢 ↦ (𝐹𝑦))) → ∃𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin) 𝐽 = 𝑠)
304, 11, 28, 29syl3anc 1363 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∃𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin) 𝐽 = 𝑠)
31 elfpw 8814 . . . . . . . 8 (𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin) ↔ (𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin))
32 simprll 775 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)))
3332sselda 3964 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) ∧ 𝑐𝑠) → 𝑐 ∈ ran (𝑦𝑢 ↦ (𝐹𝑦)))
34 simpll2 1205 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝐹:𝑋onto𝑌)
35 elssuni 4859 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐾𝑦 𝐾)
3635, 15sseqtrrdi 4015 . . . . . . . . . . . . . . . . . . . . 21 (𝑦𝐾𝑦𝑌)
377, 36syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝑦𝑌)
38 foimacnv 6625 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:𝑋onto𝑌𝑦𝑌) → (𝐹 “ (𝐹𝑦)) = 𝑦)
3934, 37, 38syl2anc 584 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → (𝐹 “ (𝐹𝑦)) = 𝑦)
40 simpr 485 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝑦𝑢)
4139, 40eqeltrd 2910 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → (𝐹 “ (𝐹𝑦)) ∈ 𝑢)
4241ralrimiva 3179 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∀𝑦𝑢 (𝐹 “ (𝐹𝑦)) ∈ 𝑢)
43 imaeq2 5918 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = (𝐹𝑦) → (𝐹𝑐) = (𝐹 “ (𝐹𝑦)))
4443eleq1d 2894 . . . . . . . . . . . . . . . . . . 19 (𝑐 = (𝐹𝑦) → ((𝐹𝑐) ∈ 𝑢 ↔ (𝐹 “ (𝐹𝑦)) ∈ 𝑢))
4524, 44ralrnmptw 6852 . . . . . . . . . . . . . . . . . 18 (∀𝑦𝑢 (𝐹𝑦) ∈ 𝐽 → (∀𝑐 ∈ ran (𝑦𝑢 ↦ (𝐹𝑦))(𝐹𝑐) ∈ 𝑢 ↔ ∀𝑦𝑢 (𝐹 “ (𝐹𝑦)) ∈ 𝑢))
4620, 45syl 17 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (∀𝑐 ∈ ran (𝑦𝑢 ↦ (𝐹𝑦))(𝐹𝑐) ∈ 𝑢 ↔ ∀𝑦𝑢 (𝐹 “ (𝐹𝑦)) ∈ 𝑢))
4742, 46mpbird 258 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∀𝑐 ∈ ran (𝑦𝑢 ↦ (𝐹𝑦))(𝐹𝑐) ∈ 𝑢)
4847adantr 481 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ∀𝑐 ∈ ran (𝑦𝑢 ↦ (𝐹𝑦))(𝐹𝑐) ∈ 𝑢)
4948r19.21bi 3205 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) ∧ 𝑐 ∈ ran (𝑦𝑢 ↦ (𝐹𝑦))) → (𝐹𝑐) ∈ 𝑢)
5033, 49syldan 591 . . . . . . . . . . . . 13 (((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) ∧ 𝑐𝑠) → (𝐹𝑐) ∈ 𝑢)
5150fmpttd 6871 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → (𝑐𝑠 ↦ (𝐹𝑐)):𝑠𝑢)
5251frnd 6514 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ran (𝑐𝑠 ↦ (𝐹𝑐)) ⊆ 𝑢)
53 simprlr 776 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑠 ∈ Fin)
54 eqid 2818 . . . . . . . . . . . . . 14 (𝑐𝑠 ↦ (𝐹𝑐)) = (𝑐𝑠 ↦ (𝐹𝑐))
5554rnmpt 5820 . . . . . . . . . . . . 13 ran (𝑐𝑠 ↦ (𝐹𝑐)) = {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)}
56 abrexfi 8812 . . . . . . . . . . . . 13 (𝑠 ∈ Fin → {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)} ∈ Fin)
5755, 56eqeltrid 2914 . . . . . . . . . . . 12 (𝑠 ∈ Fin → ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ Fin)
5853, 57syl 17 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ Fin)
59 elfpw 8814 . . . . . . . . . . 11 (ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ (𝒫 𝑢 ∩ Fin) ↔ (ran (𝑐𝑠 ↦ (𝐹𝑐)) ⊆ 𝑢 ∧ ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ Fin))
6052, 58, 59sylanbrc 583 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ (𝒫 𝑢 ∩ Fin))
6117adantr 481 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝐹: 𝐽𝑌)
6261fdmd 6516 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → dom 𝐹 = 𝐽)
63 simpll2 1205 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝐹:𝑋onto𝑌)
64 fof 6583 . . . . . . . . . . . . . 14 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
65 fdm 6515 . . . . . . . . . . . . . 14 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
6663, 64, 653syl 18 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → dom 𝐹 = 𝑋)
67 simprr 769 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝐽 = 𝑠)
6862, 66, 673eqtr3d 2861 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑋 = 𝑠)
6968imaeq2d 5922 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → (𝐹𝑋) = (𝐹 𝑠))
70 foima 6588 . . . . . . . . . . . 12 (𝐹:𝑋onto𝑌 → (𝐹𝑋) = 𝑌)
7163, 70syl 17 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → (𝐹𝑋) = 𝑌)
7250ralrimiva 3179 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ∀𝑐𝑠 (𝐹𝑐) ∈ 𝑢)
73 dfiun2g 4946 . . . . . . . . . . . . 13 (∀𝑐𝑠 (𝐹𝑐) ∈ 𝑢 𝑐𝑠 (𝐹𝑐) = {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)})
7472, 73syl 17 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑐𝑠 (𝐹𝑐) = {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)})
75 imauni 6996 . . . . . . . . . . . 12 (𝐹 𝑠) = 𝑐𝑠 (𝐹𝑐)
7655unieqi 4839 . . . . . . . . . . . 12 ran (𝑐𝑠 ↦ (𝐹𝑐)) = {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)}
7774, 75, 763eqtr4g 2878 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → (𝐹 𝑠) = ran (𝑐𝑠 ↦ (𝐹𝑐)))
7869, 71, 773eqtr3d 2861 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑌 = ran (𝑐𝑠 ↦ (𝐹𝑐)))
79 unieq 4838 . . . . . . . . . . 11 (𝑣 = ran (𝑐𝑠 ↦ (𝐹𝑐)) → 𝑣 = ran (𝑐𝑠 ↦ (𝐹𝑐)))
8079rspceeqv 3635 . . . . . . . . . 10 ((ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑌 = ran (𝑐𝑠 ↦ (𝐹𝑐))) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣)
8160, 78, 80syl2anc 584 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣)
8281expr 457 . . . . . . . 8 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ (𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin)) → ( 𝐽 = 𝑠 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
8331, 82sylan2b 593 . . . . . . 7 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin)) → ( 𝐽 = 𝑠 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
8483rexlimdva 3281 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (∃𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin) 𝐽 = 𝑠 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
8530, 84mpd 15 . . . . 5 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣)
8685expr 457 . . . 4 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢𝐾) → (𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
873, 86sylan2 592 . . 3 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢 ∈ 𝒫 𝐾) → (𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
8887ralrimiva 3179 . 2 ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑢 ∈ 𝒫 𝐾(𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
8915iscmp 21924 . 2 (𝐾 ∈ Comp ↔ (𝐾 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐾(𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣)))
902, 88, 89sylanbrc 583 1 ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  {cab 2796  wral 3135  wrex 3136  cin 3932  wss 3933  𝒫 cpw 4535   cuni 4830   ciun 4910  cmpt 5137  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551  wf 6344  ontowfo 6346  (class class class)co 7145  Fincfn 8497  Topctop 21429   Cn ccn 21760  Compccmp 21922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-fin 8501  df-top 21430  df-topon 21447  df-cn 21763  df-cmp 21923
This theorem is referenced by:  rncmp  21932  txcmpb  22180  qtopcmp  22244  cmphmph  22324
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