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Theorem cncmp 21719
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 21568 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
213ad2ant3 1116 . 2 ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3 elpwi 4435 . . . 4 (𝑢 ∈ 𝒫 𝐾𝑢𝐾)
4 simpl1 1172 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐽 ∈ Comp)
5 simpl3 1174 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐹 ∈ (𝐽 Cn 𝐾))
6 simprl 759 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝑢𝐾)
76sselda 3860 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝑦𝐾)
8 cnima 21592 . . . . . . . . . 10 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
95, 7, 8syl2an2r 673 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → (𝐹𝑦) ∈ 𝐽)
109fmpttd 6708 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝑦𝑢 ↦ (𝐹𝑦)):𝑢𝐽)
1110frnd 6356 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ran (𝑦𝑢 ↦ (𝐹𝑦)) ⊆ 𝐽)
12 simprr 761 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝑌 = 𝑢)
1312imaeq2d 5775 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝐹𝑌) = (𝐹 𝑢))
14 eqid 2780 . . . . . . . . . . 11 𝐽 = 𝐽
15 cncmp.2 . . . . . . . . . . 11 𝑌 = 𝐾
1614, 15cnf 21573 . . . . . . . . . 10 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽𝑌)
175, 16syl 17 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐹: 𝐽𝑌)
18 fimacnv 6670 . . . . . . . . 9 (𝐹: 𝐽𝑌 → (𝐹𝑌) = 𝐽)
1917, 18syl 17 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝐹𝑌) = 𝐽)
209ralrimiva 3134 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∀𝑦𝑢 (𝐹𝑦) ∈ 𝐽)
21 dfiun2g 4830 . . . . . . . . . 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 6836 . . . . . . . . 9 (𝐹 𝑢) = 𝑦𝑢 (𝐹𝑦)
24 eqid 2780 . . . . . . . . . . 11 (𝑦𝑢 ↦ (𝐹𝑦)) = (𝑦𝑢 ↦ (𝐹𝑦))
2524rnmpt 5675 . . . . . . . . . 10 ran (𝑦𝑢 ↦ (𝐹𝑦)) = {𝑥 ∣ ∃𝑦𝑢 𝑥 = (𝐹𝑦)}
2625unieqi 4726 . . . . . . . . 9 ran (𝑦𝑢 ↦ (𝐹𝑦)) = {𝑥 ∣ ∃𝑦𝑢 𝑥 = (𝐹𝑦)}
2722, 23, 263eqtr4g 2841 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝐹 𝑢) = ran (𝑦𝑢 ↦ (𝐹𝑦)))
2813, 19, 273eqtr3d 2824 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐽 = ran (𝑦𝑢 ↦ (𝐹𝑦)))
2914cmpcov 21716 . . . . . . 7 ((𝐽 ∈ Comp ∧ ran (𝑦𝑢 ↦ (𝐹𝑦)) ⊆ 𝐽 𝐽 = ran (𝑦𝑢 ↦ (𝐹𝑦))) → ∃𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin) 𝐽 = 𝑠)
304, 11, 28, 29syl3anc 1352 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∃𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin) 𝐽 = 𝑠)
31 elfpw 8627 . . . . . . . 8 (𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin) ↔ (𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin))
32 simprll 767 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)))
3332sselda 3860 . . . . . . . . . . . . . 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 1194 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝐹:𝑋onto𝑌)
35 elssuni 4746 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐾𝑦 𝐾)
3635, 15syl6sseqr 3910 . . . . . . . . . . . . . . . . . . . . 21 (𝑦𝐾𝑦𝑌)
377, 36syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝑦𝑌)
38 foimacnv 6466 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:𝑋onto𝑌𝑦𝑌) → (𝐹 “ (𝐹𝑦)) = 𝑦)
3934, 37, 38syl2anc 576 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → (𝐹 “ (𝐹𝑦)) = 𝑦)
40 simpr 477 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝑦𝑢)
4139, 40eqeltrd 2868 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → (𝐹 “ (𝐹𝑦)) ∈ 𝑢)
4241ralrimiva 3134 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∀𝑦𝑢 (𝐹 “ (𝐹𝑦)) ∈ 𝑢)
43 imaeq2 5771 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = (𝐹𝑦) → (𝐹𝑐) = (𝐹 “ (𝐹𝑦)))
4443eleq1d 2852 . . . . . . . . . . . . . . . . . . 19 (𝑐 = (𝐹𝑦) → ((𝐹𝑐) ∈ 𝑢 ↔ (𝐹 “ (𝐹𝑦)) ∈ 𝑢))
4524, 44ralrnmpt 6691 . . . . . . . . . . . . . . . . . 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 249 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∀𝑐 ∈ ran (𝑦𝑢 ↦ (𝐹𝑦))(𝐹𝑐) ∈ 𝑢)
4847adantr 473 . . . . . . . . . . . . . . 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 3160 . . . . . . . . . . . . . 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 583 . . . . . . . . . . . . 13 (((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) ∧ 𝑐𝑠) → (𝐹𝑐) ∈ 𝑢)
5150fmpttd 6708 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → (𝑐𝑠 ↦ (𝐹𝑐)):𝑠𝑢)
5251frnd 6356 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ran (𝑐𝑠 ↦ (𝐹𝑐)) ⊆ 𝑢)
53 simprlr 768 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑠 ∈ Fin)
54 eqid 2780 . . . . . . . . . . . . . 14 (𝑐𝑠 ↦ (𝐹𝑐)) = (𝑐𝑠 ↦ (𝐹𝑐))
5554rnmpt 5675 . . . . . . . . . . . . 13 ran (𝑐𝑠 ↦ (𝐹𝑐)) = {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)}
56 abrexfi 8625 . . . . . . . . . . . . 13 (𝑠 ∈ Fin → {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)} ∈ Fin)
5755, 56syl5eqel 2872 . . . . . . . . . . . 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 8627 . . . . . . . . . . 11 (ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ (𝒫 𝑢 ∩ Fin) ↔ (ran (𝑐𝑠 ↦ (𝐹𝑐)) ⊆ 𝑢 ∧ ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ Fin))
6052, 58, 59sylanbrc 575 . . . . . . . . . 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 473 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝐹: 𝐽𝑌)
6261fdmd 6358 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → dom 𝐹 = 𝐽)
63 simpll2 1194 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝐹:𝑋onto𝑌)
64 fof 6424 . . . . . . . . . . . . . 14 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
65 fdm 6357 . . . . . . . . . . . . . 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 761 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝐽 = 𝑠)
6862, 66, 673eqtr3d 2824 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑋 = 𝑠)
6968imaeq2d 5775 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → (𝐹𝑋) = (𝐹 𝑠))
70 foima 6429 . . . . . . . . . . . 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 3134 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ∀𝑐𝑠 (𝐹𝑐) ∈ 𝑢)
73 dfiun2g 4830 . . . . . . . . . . . . 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 6836 . . . . . . . . . . . 12 (𝐹 𝑠) = 𝑐𝑠 (𝐹𝑐)
7655unieqi 4726 . . . . . . . . . . . 12 ran (𝑐𝑠 ↦ (𝐹𝑐)) = {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)}
7774, 75, 763eqtr4g 2841 . . . . . . . . . . 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 2824 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑌 = ran (𝑐𝑠 ↦ (𝐹𝑐)))
79 unieq 4725 . . . . . . . . . . 11 (𝑣 = ran (𝑐𝑠 ↦ (𝐹𝑐)) → 𝑣 = ran (𝑐𝑠 ↦ (𝐹𝑐)))
8079rspceeqv 3555 . . . . . . . . . 10 ((ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑌 = ran (𝑐𝑠 ↦ (𝐹𝑐))) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣)
8160, 78, 80syl2anc 576 . . . . . . . . 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 449 . . . . . . . 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 585 . . . . . . 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 3231 . . . . . 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 449 . . . 4 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢𝐾) → (𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
873, 86sylan2 584 . . 3 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢 ∈ 𝒫 𝐾) → (𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
8887ralrimiva 3134 . 2 ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑢 ∈ 𝒫 𝐾(𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
8915iscmp 21715 . 2 (𝐾 ∈ Comp ↔ (𝐾 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐾(𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣)))
902, 88, 89sylanbrc 575 1 ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1069   = wceq 1508  wcel 2051  {cab 2760  wral 3090  wrex 3091  cin 3830  wss 3831  𝒫 cpw 4425   cuni 4717   ciun 4797  cmpt 5013  ccnv 5410  dom cdm 5411  ran crn 5412  cima 5414  wf 6189  ontowfo 6191  (class class class)co 6982  Fincfn 8312  Topctop 21220   Cn ccn 21551  Compccmp 21713
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-1st 7507  df-2nd 7508  df-wrecs 7756  df-recs 7818  df-rdg 7856  df-1o 7911  df-oadd 7915  df-er 8095  df-map 8214  df-en 8313  df-dom 8314  df-fin 8316  df-top 21221  df-topon 21238  df-cn 21554  df-cmp 21714
This theorem is referenced by:  rncmp  21723  txcmpb  21971  qtopcmp  22035  cmphmph  22115
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