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Theorem cncmp 22001
 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 21850 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
213ad2ant3 1132 . 2 ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3 elpwi 4509 . . . 4 (𝑢 ∈ 𝒫 𝐾𝑢𝐾)
4 simpl1 1188 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐽 ∈ Comp)
5 simpl3 1190 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐹 ∈ (𝐽 Cn 𝐾))
6 simprl 770 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝑢𝐾)
76sselda 3918 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝑦𝐾)
8 cnima 21874 . . . . . . . . . 10 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
95, 7, 8syl2an2r 684 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → (𝐹𝑦) ∈ 𝐽)
109fmpttd 6860 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝑦𝑢 ↦ (𝐹𝑦)):𝑢𝐽)
1110frnd 6498 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ran (𝑦𝑢 ↦ (𝐹𝑦)) ⊆ 𝐽)
12 simprr 772 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝑌 = 𝑢)
1312imaeq2d 5900 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝐹𝑌) = (𝐹 𝑢))
14 eqid 2801 . . . . . . . . . . 11 𝐽 = 𝐽
15 cncmp.2 . . . . . . . . . . 11 𝑌 = 𝐾
1614, 15cnf 21855 . . . . . . . . . 10 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽𝑌)
175, 16syl 17 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐹: 𝐽𝑌)
18 fimacnv 6820 . . . . . . . . 9 (𝐹: 𝐽𝑌 → (𝐹𝑌) = 𝐽)
1917, 18syl 17 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝐹𝑌) = 𝐽)
209ralrimiva 3152 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∀𝑦𝑢 (𝐹𝑦) ∈ 𝐽)
21 dfiun2g 4920 . . . . . . . . . 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 6987 . . . . . . . . 9 (𝐹 𝑢) = 𝑦𝑢 (𝐹𝑦)
24 eqid 2801 . . . . . . . . . . 11 (𝑦𝑢 ↦ (𝐹𝑦)) = (𝑦𝑢 ↦ (𝐹𝑦))
2524rnmpt 5795 . . . . . . . . . 10 ran (𝑦𝑢 ↦ (𝐹𝑦)) = {𝑥 ∣ ∃𝑦𝑢 𝑥 = (𝐹𝑦)}
2625unieqi 4816 . . . . . . . . 9 ran (𝑦𝑢 ↦ (𝐹𝑦)) = {𝑥 ∣ ∃𝑦𝑢 𝑥 = (𝐹𝑦)}
2722, 23, 263eqtr4g 2861 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝐹 𝑢) = ran (𝑦𝑢 ↦ (𝐹𝑦)))
2813, 19, 273eqtr3d 2844 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐽 = ran (𝑦𝑢 ↦ (𝐹𝑦)))
2914cmpcov 21998 . . . . . . 7 ((𝐽 ∈ Comp ∧ ran (𝑦𝑢 ↦ (𝐹𝑦)) ⊆ 𝐽 𝐽 = ran (𝑦𝑢 ↦ (𝐹𝑦))) → ∃𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin) 𝐽 = 𝑠)
304, 11, 28, 29syl3anc 1368 . . . . . 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 778 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)))
3332sselda 3918 . . . . . . . . . . . . . 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 1210 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝐹:𝑋onto𝑌)
35 elssuni 4833 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐾𝑦 𝐾)
3635, 15sseqtrrdi 3969 . . . . . . . . . . . . . . . . . . . . 21 (𝑦𝐾𝑦𝑌)
377, 36syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝑦𝑌)
38 foimacnv 6611 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:𝑋onto𝑌𝑦𝑌) → (𝐹 “ (𝐹𝑦)) = 𝑦)
3934, 37, 38syl2anc 587 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → (𝐹 “ (𝐹𝑦)) = 𝑦)
40 simpr 488 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝑦𝑢)
4139, 40eqeltrd 2893 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → (𝐹 “ (𝐹𝑦)) ∈ 𝑢)
4241ralrimiva 3152 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∀𝑦𝑢 (𝐹 “ (𝐹𝑦)) ∈ 𝑢)
43 imaeq2 5896 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = (𝐹𝑦) → (𝐹𝑐) = (𝐹 “ (𝐹𝑦)))
4443eleq1d 2877 . . . . . . . . . . . . . . . . . . 19 (𝑐 = (𝐹𝑦) → ((𝐹𝑐) ∈ 𝑢 ↔ (𝐹 “ (𝐹𝑦)) ∈ 𝑢))
4524, 44ralrnmptw 6841 . . . . . . . . . . . . . . . . . 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 260 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∀𝑐 ∈ ran (𝑦𝑢 ↦ (𝐹𝑦))(𝐹𝑐) ∈ 𝑢)
4847adantr 484 . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . 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 594 . . . . . . . . . . . . 13 (((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) ∧ 𝑐𝑠) → (𝐹𝑐) ∈ 𝑢)
5150fmpttd 6860 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → (𝑐𝑠 ↦ (𝐹𝑐)):𝑠𝑢)
5251frnd 6498 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ran (𝑐𝑠 ↦ (𝐹𝑐)) ⊆ 𝑢)
53 simprlr 779 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑠 ∈ Fin)
54 eqid 2801 . . . . . . . . . . . . . 14 (𝑐𝑠 ↦ (𝐹𝑐)) = (𝑐𝑠 ↦ (𝐹𝑐))
5554rnmpt 5795 . . . . . . . . . . . . 13 ran (𝑐𝑠 ↦ (𝐹𝑐)) = {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)}
56 abrexfi 8812 . . . . . . . . . . . . 13 (𝑠 ∈ Fin → {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)} ∈ Fin)
5755, 56eqeltrid 2897 . . . . . . . . . . . 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 586 . . . . . . . . . 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 484 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝐹: 𝐽𝑌)
6261fdmd 6501 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → dom 𝐹 = 𝐽)
63 simpll2 1210 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝐹:𝑋onto𝑌)
64 fof 6569 . . . . . . . . . . . . . 14 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
65 fdm 6499 . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝐽 = 𝑠)
6862, 66, 673eqtr3d 2844 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑋 = 𝑠)
6968imaeq2d 5900 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → (𝐹𝑋) = (𝐹 𝑠))
70 foima 6574 . . . . . . . . . . . 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 3152 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ∀𝑐𝑠 (𝐹𝑐) ∈ 𝑢)
73 dfiun2g 4920 . . . . . . . . . . . . 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 6987 . . . . . . . . . . . 12 (𝐹 𝑠) = 𝑐𝑠 (𝐹𝑐)
7655unieqi 4816 . . . . . . . . . . . 12 ran (𝑐𝑠 ↦ (𝐹𝑐)) = {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)}
7774, 75, 763eqtr4g 2861 . . . . . . . . . . 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 2844 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑌 = ran (𝑐𝑠 ↦ (𝐹𝑐)))
79 unieq 4814 . . . . . . . . . . 11 (𝑣 = ran (𝑐𝑠 ↦ (𝐹𝑐)) → 𝑣 = ran (𝑐𝑠 ↦ (𝐹𝑐)))
8079rspceeqv 3589 . . . . . . . . . 10 ((ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑌 = ran (𝑐𝑠 ↦ (𝐹𝑐))) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣)
8160, 78, 80syl2anc 587 . . . . . . . . 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 460 . . . . . . . 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 596 . . . . . . 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 3246 . . . . . 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 460 . . . 4 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢𝐾) → (𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
873, 86sylan2 595 . . 3 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢 ∈ 𝒫 𝐾) → (𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
8887ralrimiva 3152 . 2 ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑢 ∈ 𝒫 𝐾(𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
8915iscmp 21997 . 2 (𝐾 ∈ Comp ↔ (𝐾 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐾(𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣)))
902, 88, 89sylanbrc 586 1 ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2112  {cab 2779  ∀wral 3109  ∃wrex 3110   ∩ cin 3883   ⊆ wss 3884  𝒫 cpw 4500  ∪ cuni 4803  ∪ ciun 4884   ↦ cmpt 5113  ◡ccnv 5522  dom cdm 5523  ran crn 5524   “ cima 5526  ⟶wf 6324  –onto→wfo 6326  (class class class)co 7139  Fincfn 8496  Topctop 21502   Cn ccn 21833  Compccmp 21995 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-int 4842  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-om 7565  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-1o 8089  df-oadd 8093  df-er 8276  df-map 8395  df-en 8497  df-dom 8498  df-fin 8500  df-top 21503  df-topon 21520  df-cn 21836  df-cmp 21996 This theorem is referenced by:  rncmp  22005  txcmpb  22253  qtopcmp  22317  cmphmph  22397
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