Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  cncmp Structured version   Visualization version   GIF version

Theorem cncmp 21005
 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 20855 . . 3 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
213ad2ant3 1077 . 2 ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Top)
3 elpwi 4117 . . . 4 (𝑢 ∈ 𝒫 𝐾𝑢𝐾)
4 simpl1 1057 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐽 ∈ Comp)
5 simprl 790 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝑢𝐾)
65sselda 3568 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝑦𝐾)
7 simpl3 1059 . . . . . . . . . . 11 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐹 ∈ (𝐽 Cn 𝐾))
8 cnima 20879 . . . . . . . . . . 11 ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
97, 8sylan 487 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝐾) → (𝐹𝑦) ∈ 𝐽)
106, 9syldan 486 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → (𝐹𝑦) ∈ 𝐽)
11 eqid 2610 . . . . . . . . 9 (𝑦𝑢 ↦ (𝐹𝑦)) = (𝑦𝑢 ↦ (𝐹𝑦))
1210, 11fmptd 6292 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝑦𝑢 ↦ (𝐹𝑦)):𝑢𝐽)
13 frn 5966 . . . . . . . 8 ((𝑦𝑢 ↦ (𝐹𝑦)):𝑢𝐽 → ran (𝑦𝑢 ↦ (𝐹𝑦)) ⊆ 𝐽)
1412, 13syl 17 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ran (𝑦𝑢 ↦ (𝐹𝑦)) ⊆ 𝐽)
15 simprr 792 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝑌 = 𝑢)
1615imaeq2d 5385 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝐹𝑌) = (𝐹 𝑢))
17 eqid 2610 . . . . . . . . . . 11 𝐽 = 𝐽
18 cncmp.2 . . . . . . . . . . 11 𝑌 = 𝐾
1917, 18cnf 20860 . . . . . . . . . 10 (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐹: 𝐽𝑌)
207, 19syl 17 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐹: 𝐽𝑌)
21 fimacnv 6255 . . . . . . . . 9 (𝐹: 𝐽𝑌 → (𝐹𝑌) = 𝐽)
2220, 21syl 17 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝐹𝑌) = 𝐽)
2310ralrimiva 2949 . . . . . . . . . 10 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∀𝑦𝑢 (𝐹𝑦) ∈ 𝐽)
24 dfiun2g 4488 . . . . . . . . . 10 (∀𝑦𝑢 (𝐹𝑦) ∈ 𝐽 𝑦𝑢 (𝐹𝑦) = {𝑥 ∣ ∃𝑦𝑢 𝑥 = (𝐹𝑦)})
2523, 24syl 17 . . . . . . . . 9 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝑦𝑢 (𝐹𝑦) = {𝑥 ∣ ∃𝑦𝑢 𝑥 = (𝐹𝑦)})
26 imauni 6408 . . . . . . . . 9 (𝐹 𝑢) = 𝑦𝑢 (𝐹𝑦)
2711rnmpt 5292 . . . . . . . . . 10 ran (𝑦𝑢 ↦ (𝐹𝑦)) = {𝑥 ∣ ∃𝑦𝑢 𝑥 = (𝐹𝑦)}
2827unieqi 4381 . . . . . . . . 9 ran (𝑦𝑢 ↦ (𝐹𝑦)) = {𝑥 ∣ ∃𝑦𝑢 𝑥 = (𝐹𝑦)}
2925, 26, 283eqtr4g 2669 . . . . . . . 8 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (𝐹 𝑢) = ran (𝑦𝑢 ↦ (𝐹𝑦)))
3016, 22, 293eqtr3d 2652 . . . . . . 7 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → 𝐽 = ran (𝑦𝑢 ↦ (𝐹𝑦)))
3117cmpcov 21002 . . . . . . 7 ((𝐽 ∈ Comp ∧ ran (𝑦𝑢 ↦ (𝐹𝑦)) ⊆ 𝐽 𝐽 = ran (𝑦𝑢 ↦ (𝐹𝑦))) → ∃𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin) 𝐽 = 𝑠)
324, 14, 30, 31syl3anc 1318 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∃𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin) 𝐽 = 𝑠)
33 elfpw 8151 . . . . . . . 8 (𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin) ↔ (𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin))
34 simprll 798 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)))
3534sselda 3568 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) ∧ 𝑐𝑠) → 𝑐 ∈ ran (𝑦𝑢 ↦ (𝐹𝑦)))
36 simpll2 1094 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝐹:𝑋onto𝑌)
37 elssuni 4403 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝐾𝑦 𝐾)
3837, 18syl6sseqr 3615 . . . . . . . . . . . . . . . . . . . . 21 (𝑦𝐾𝑦𝑌)
396, 38syl 17 . . . . . . . . . . . . . . . . . . . 20 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝑦𝑌)
40 foimacnv 6067 . . . . . . . . . . . . . . . . . . . 20 ((𝐹:𝑋onto𝑌𝑦𝑌) → (𝐹 “ (𝐹𝑦)) = 𝑦)
4136, 39, 40syl2anc 691 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → (𝐹 “ (𝐹𝑦)) = 𝑦)
42 simpr 476 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → 𝑦𝑢)
4341, 42eqeltrd 2688 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑦𝑢) → (𝐹 “ (𝐹𝑦)) ∈ 𝑢)
4443ralrimiva 2949 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∀𝑦𝑢 (𝐹 “ (𝐹𝑦)) ∈ 𝑢)
45 imaeq2 5381 . . . . . . . . . . . . . . . . . . . 20 (𝑐 = (𝐹𝑦) → (𝐹𝑐) = (𝐹 “ (𝐹𝑦)))
4645eleq1d 2672 . . . . . . . . . . . . . . . . . . 19 (𝑐 = (𝐹𝑦) → ((𝐹𝑐) ∈ 𝑢 ↔ (𝐹 “ (𝐹𝑦)) ∈ 𝑢))
4711, 46ralrnmpt 6276 . . . . . . . . . . . . . . . . . 18 (∀𝑦𝑢 (𝐹𝑦) ∈ 𝐽 → (∀𝑐 ∈ ran (𝑦𝑢 ↦ (𝐹𝑦))(𝐹𝑐) ∈ 𝑢 ↔ ∀𝑦𝑢 (𝐹 “ (𝐹𝑦)) ∈ 𝑢))
4823, 47syl 17 . . . . . . . . . . . . . . . . 17 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (∀𝑐 ∈ ran (𝑦𝑢 ↦ (𝐹𝑦))(𝐹𝑐) ∈ 𝑢 ↔ ∀𝑦𝑢 (𝐹 “ (𝐹𝑦)) ∈ 𝑢))
4944, 48mpbird 246 . . . . . . . . . . . . . . . 16 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∀𝑐 ∈ ran (𝑦𝑢 ↦ (𝐹𝑦))(𝐹𝑐) ∈ 𝑢)
5049adantr 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)
5150r19.21bi 2916 . . . . . . . . . . . . . 14 (((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) ∧ 𝑐 ∈ ran (𝑦𝑢 ↦ (𝐹𝑦))) → (𝐹𝑐) ∈ 𝑢)
5235, 51syldan 486 . . . . . . . . . . . . 13 (((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) ∧ 𝑐𝑠) → (𝐹𝑐) ∈ 𝑢)
53 eqid 2610 . . . . . . . . . . . . 13 (𝑐𝑠 ↦ (𝐹𝑐)) = (𝑐𝑠 ↦ (𝐹𝑐))
5452, 53fmptd 6292 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → (𝑐𝑠 ↦ (𝐹𝑐)):𝑠𝑢)
55 frn 5966 . . . . . . . . . . . 12 ((𝑐𝑠 ↦ (𝐹𝑐)):𝑠𝑢 → ran (𝑐𝑠 ↦ (𝐹𝑐)) ⊆ 𝑢)
5654, 55syl 17 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ran (𝑐𝑠 ↦ (𝐹𝑐)) ⊆ 𝑢)
57 simprlr 799 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑠 ∈ Fin)
5853rnmpt 5292 . . . . . . . . . . . . 13 ran (𝑐𝑠 ↦ (𝐹𝑐)) = {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)}
59 abrexfi 8149 . . . . . . . . . . . . 13 (𝑠 ∈ Fin → {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)} ∈ Fin)
6058, 59syl5eqel 2692 . . . . . . . . . . . 12 (𝑠 ∈ Fin → ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ Fin)
6157, 60syl 17 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ Fin)
62 elfpw 8151 . . . . . . . . . . 11 (ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ (𝒫 𝑢 ∩ Fin) ↔ (ran (𝑐𝑠 ↦ (𝐹𝑐)) ⊆ 𝑢 ∧ ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ Fin))
6356, 61, 62sylanbrc 695 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ (𝒫 𝑢 ∩ Fin))
6420adantr 480 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝐹: 𝐽𝑌)
65 fdm 5964 . . . . . . . . . . . . . 14 (𝐹: 𝐽𝑌 → dom 𝐹 = 𝐽)
6664, 65syl 17 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → dom 𝐹 = 𝐽)
67 simpll2 1094 . . . . . . . . . . . . . 14 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝐹:𝑋onto𝑌)
68 fof 6028 . . . . . . . . . . . . . 14 (𝐹:𝑋onto𝑌𝐹:𝑋𝑌)
69 fdm 5964 . . . . . . . . . . . . . 14 (𝐹:𝑋𝑌 → dom 𝐹 = 𝑋)
7067, 68, 693syl 18 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → dom 𝐹 = 𝑋)
71 simprr 792 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝐽 = 𝑠)
7266, 70, 713eqtr3d 2652 . . . . . . . . . . . 12 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑋 = 𝑠)
7372imaeq2d 5385 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → (𝐹𝑋) = (𝐹 𝑠))
74 foima 6033 . . . . . . . . . . . 12 (𝐹:𝑋onto𝑌 → (𝐹𝑋) = 𝑌)
7567, 74syl 17 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → (𝐹𝑋) = 𝑌)
7652ralrimiva 2949 . . . . . . . . . . . . 13 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ∀𝑐𝑠 (𝐹𝑐) ∈ 𝑢)
77 dfiun2g 4488 . . . . . . . . . . . . 13 (∀𝑐𝑠 (𝐹𝑐) ∈ 𝑢 𝑐𝑠 (𝐹𝑐) = {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)})
7876, 77syl 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)})
79 imauni 6408 . . . . . . . . . . . 12 (𝐹 𝑠) = 𝑐𝑠 (𝐹𝑐)
8058unieqi 4381 . . . . . . . . . . . 12 ran (𝑐𝑠 ↦ (𝐹𝑐)) = {𝑑 ∣ ∃𝑐𝑠 𝑑 = (𝐹𝑐)}
8178, 79, 803eqtr4g 2669 . . . . . . . . . . 11 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → (𝐹 𝑠) = ran (𝑐𝑠 ↦ (𝐹𝑐)))
8273, 75, 813eqtr3d 2652 . . . . . . . . . 10 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → 𝑌 = ran (𝑐𝑠 ↦ (𝐹𝑐)))
83 unieq 4380 . . . . . . . . . . . 12 (𝑣 = ran (𝑐𝑠 ↦ (𝐹𝑐)) → 𝑣 = ran (𝑐𝑠 ↦ (𝐹𝑐)))
8483eqeq2d 2620 . . . . . . . . . . 11 (𝑣 = ran (𝑐𝑠 ↦ (𝐹𝑐)) → (𝑌 = 𝑣𝑌 = ran (𝑐𝑠 ↦ (𝐹𝑐))))
8584rspcev 3282 . . . . . . . . . 10 ((ran (𝑐𝑠 ↦ (𝐹𝑐)) ∈ (𝒫 𝑢 ∩ Fin) ∧ 𝑌 = ran (𝑐𝑠 ↦ (𝐹𝑐))) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣)
8663, 82, 85syl2anc 691 . . . . . . . . 9 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ ((𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin) ∧ 𝐽 = 𝑠)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣)
8786expr 641 . . . . . . . 8 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ (𝑠 ⊆ ran (𝑦𝑢 ↦ (𝐹𝑦)) ∧ 𝑠 ∈ Fin)) → ( 𝐽 = 𝑠 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
8833, 87sylan2b 491 . . . . . . 7 ((((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) ∧ 𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin)) → ( 𝐽 = 𝑠 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
8988rexlimdva 3013 . . . . . 6 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → (∃𝑠 ∈ (𝒫 ran (𝑦𝑢 ↦ (𝐹𝑦)) ∩ Fin) 𝐽 = 𝑠 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
9032, 89mpd 15 . . . . 5 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ (𝑢𝐾𝑌 = 𝑢)) → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣)
9190expr 641 . . . 4 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢𝐾) → (𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
923, 91sylan2 490 . . 3 (((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) ∧ 𝑢 ∈ 𝒫 𝐾) → (𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
9392ralrimiva 2949 . 2 ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → ∀𝑢 ∈ 𝒫 𝐾(𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣))
9418iscmp 21001 . 2 (𝐾 ∈ Comp ↔ (𝐾 ∈ Top ∧ ∀𝑢 ∈ 𝒫 𝐾(𝑌 = 𝑢 → ∃𝑣 ∈ (𝒫 𝑢 ∩ Fin)𝑌 = 𝑣)))
952, 93, 94sylanbrc 695 1 ((𝐽 ∈ Comp ∧ 𝐹:𝑋onto𝑌𝐹 ∈ (𝐽 Cn 𝐾)) → 𝐾 ∈ Comp)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ∪ cuni 4372  ∪ ciun 4455   ↦ cmpt 4643  ◡ccnv 5037  dom cdm 5038  ran crn 5039   “ cima 5041  ⟶wf 5800  –onto→wfo 5802  (class class class)co 6549  Fincfn 7841  Topctop 20517   Cn ccn 20838  Compccmp 20999 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-oadd 7451  df-er 7629  df-map 7746  df-en 7842  df-dom 7843  df-fin 7845  df-top 20521  df-topon 20523  df-cn 20841  df-cmp 21000 This theorem is referenced by:  rncmp  21009  txcmpb  21257  qtopcmp  21321  cmphmph  21401
 Copyright terms: Public domain W3C validator