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Theorem cncmp 23416
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 23265 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
213ad2ant3 1134 . 2 ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3 elpwi 4612 . . . 4 (𝑢 ∈ 𝒫 𝐾𝑢𝐾)
4 simpl1 1190 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐽 ∈ Comp)
5 simpl3 1192 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐹 ∈ (𝐽 Cn 𝐾))
6 simprl 771 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝑢𝐾)
76sselda 3995 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝑦𝐾)
8 cnima 23289 . . . . . . . . . 10 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
95, 7, 8syl2an2r 685 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → (𝐹𝑦) ∈ 𝐽)
109fmpttd 7135 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝑦𝑢 ↦ (𝐹𝑦)):𝑢𝐽)
1110frnd 6745 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ran (𝑦𝑢 ↦ (𝐹𝑦)) ⊆ 𝐽)
12 simprr 773 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝑌 = 𝑢)
1312imaeq2d 6080 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝐹𝑌) = (𝐹 𝑢))
14 eqid 2735 . . . . . . . . . . 11 𝐽 = 𝐽
15 cncmp.2 . . . . . . . . . . 11 𝑌 = 𝐾
1614, 15cnf 23270 . . . . . . . . . 10 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽𝑌)
175, 16syl 17 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐹: 𝐽𝑌)
18 fimacnv 6759 . . . . . . . . 9 (𝐹: 𝐽𝑌 → (𝐹𝑌) = 𝐽)
1917, 18syl 17 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝐹𝑌) = 𝐽)
209ralrimiva 3144 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∀𝑦𝑢 (𝐹𝑦) ∈ 𝐽)
21 dfiun2g 5035 . . . . . . . . . 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 7266 . . . . . . . . 9 (𝐹 𝑢) = 𝑦𝑢 (𝐹𝑦)
24 eqid 2735 . . . . . . . . . . 11 (𝑦𝑢 ↦ (𝐹𝑦)) = (𝑦𝑢 ↦ (𝐹𝑦))
2524rnmpt 5971 . . . . . . . . . 10 ran (𝑦𝑢 ↦ (𝐹𝑦)) = {𝑥 ∣ ∃𝑦𝑢 𝑥 = (𝐹𝑦)}
2625unieqi 4924 . . . . . . . . 9 ran (𝑦𝑢 ↦ (𝐹𝑦)) = {𝑥 ∣ ∃𝑦𝑢 𝑥 = (𝐹𝑦)}
2722, 23, 263eqtr4g 2800 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝐹 𝑢) = ran (𝑦𝑢 ↦ (𝐹𝑦)))
2813, 19, 273eqtr3d 2783 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐽 = ran (𝑦𝑢 ↦ (𝐹𝑦)))
2914cmpcov 23413 . . . . . . 7 ((𝐽 ∈ Comp ∧ ran (𝑦𝑢 ↦ (𝐹𝑦)) ⊆ 𝐽 𝐽 = ran (𝑦𝑢 ↦ (𝐹𝑦))) → ∃𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin) 𝐽 = 𝑠)
304, 11, 28, 29syl3anc 1370 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∃𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin) 𝐽 = 𝑠)
31 elfpw 9392 . . . . . . . 8 (𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin) ↔ (𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin))
32 simprll 779 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)))
3332sselda 3995 . . . . . . . . . . . . . 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 1212 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝐹:𝑋onto𝑌)
35 elssuni 4942 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐾𝑦 𝐾)
3635, 15sseqtrrdi 4047 . . . . . . . . . . . . . . . . . . . . 21 (𝑦𝐾𝑦𝑌)
377, 36syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝑦𝑌)
38 foimacnv 6866 . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝑦𝑢)
4139, 40eqeltrd 2839 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → (𝐹 “ (𝐹𝑦)) ∈ 𝑢)
4241ralrimiva 3144 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∀𝑦𝑢 (𝐹 “ (𝐹𝑦)) ∈ 𝑢)
43 imaeq2 6076 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = (𝐹𝑦) → (𝐹𝑐) = (𝐹 “ (𝐹𝑦)))
4443eleq1d 2824 . . . . . . . . . . . . . . . . . . 19 (𝑐 = (𝐹𝑦) → ((𝐹𝑐) ∈ 𝑢 ↔ (𝐹 “ (𝐹𝑦)) ∈ 𝑢))
4524, 44ralrnmptw 7114 . . . . . . . . . . . . . . . . . 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 257 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∀𝑐 ∈ ran (𝑦𝑢 ↦ (𝐹𝑦))(𝐹𝑐) ∈ 𝑢)
4847adantr 480 . . . . . . . . . . . . . . 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 3249 . . . . . . . . . . . . . 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 7135 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → (𝑐𝑠 ↦ (𝐹𝑐)):𝑠𝑢)
5251frnd 6745 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ran (𝑐𝑠 ↦ (𝐹𝑐)) ⊆ 𝑢)
53 simprlr 780 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑠 ∈ Fin)
54 eqid 2735 . . . . . . . . . . . . . 14 (𝑐𝑠 ↦ (𝐹𝑐)) = (𝑐𝑠 ↦ (𝐹𝑐))
5554rnmpt 5971 . . . . . . . . . . . . 13 ran (𝑐𝑠 ↦ (𝐹𝑐)) = {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)}
56 abrexfi 9390 . . . . . . . . . . . . 13 (𝑠 ∈ Fin → {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)} ∈ Fin)
5755, 56eqeltrid 2843 . . . . . . . . . . . 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 9392 . . . . . . . . . . 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 480 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝐹: 𝐽𝑌)
6261fdmd 6747 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → dom 𝐹 = 𝐽)
63 simpll2 1212 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝐹:𝑋onto𝑌)
64 fof 6821 . . . . . . . . . . . . . 14 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
65 fdm 6746 . . . . . . . . . . . . . 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 773 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝐽 = 𝑠)
6862, 66, 673eqtr3d 2783 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑋 = 𝑠)
6968imaeq2d 6080 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → (𝐹𝑋) = (𝐹 𝑠))
70 foima 6826 . . . . . . . . . . . 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 3144 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ∀𝑐𝑠 (𝐹𝑐) ∈ 𝑢)
73 dfiun2g 5035 . . . . . . . . . . . . 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 7266 . . . . . . . . . . . 12 (𝐹 𝑠) = 𝑐𝑠 (𝐹𝑐)
7655unieqi 4924 . . . . . . . . . . . 12 ran (𝑐𝑠 ↦ (𝐹𝑐)) = {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)}
7774, 75, 763eqtr4g 2800 . . . . . . . . . . 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 2783 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑌 = ran (𝑐𝑠 ↦ (𝐹𝑐)))
79 unieq 4923 . . . . . . . . . . 11 (𝑣 = ran (𝑐𝑠 ↦ (𝐹𝑐)) → 𝑣 = ran (𝑐𝑠 ↦ (𝐹𝑐)))
8079rspceeqv 3645 . . . . . . . . . 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 456 . . . . . . . 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 594 . . . . . . 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 3153 . . . . . 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 456 . . . 4 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢𝐾) → (𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
873, 86sylan2 593 . . 3 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢 ∈ 𝒫 𝐾) → (𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
8887ralrimiva 3144 . 2 ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑢 ∈ 𝒫 𝐾(𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
8915iscmp 23412 . 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 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  {cab 2712  wral 3059  wrex 3068  cin 3962  wss 3963  𝒫 cpw 4605   cuni 4912   ciun 4996  cmpt 5231  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692  wf 6559  ontowfo 6561  (class class class)co 7431  Fincfn 8984  Topctop 22915   Cn ccn 23248  Compccmp 23410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-1o 8505  df-map 8867  df-en 8985  df-dom 8986  df-fin 8988  df-top 22916  df-topon 22933  df-cn 23251  df-cmp 23411
This theorem is referenced by:  rncmp  23420  txcmpb  23668  qtopcmp  23732  cmphmph  23812
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