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

Theorem cnextrel 22599
Description: In the general case, a continuous extension is a relation. (Contributed by Thierry Arnoux, 20-Dec-2017.)
Hypotheses
Ref Expression
cnextfrel.1 𝐶 = 𝐽
cnextfrel.2 𝐵 = 𝐾
Assertion
Ref Expression
cnextrel (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → Rel ((𝐽CnExt𝐾)‘𝐹))

Proof of Theorem cnextrel
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 relxp 5566 . . . 4 Rel ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
21rgenw 3147 . . 3 𝑥 ∈ ((cls‘𝐽)‘𝐴)Rel ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
3 reliun 5682 . . 3 (Rel 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ ∀𝑥 ∈ ((cls‘𝐽)‘𝐴)Rel ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
42, 3mpbir 232 . 2 Rel 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
5 cnextfrel.1 . . . 4 𝐶 = 𝐽
6 cnextfrel.2 . . . 4 𝐵 = 𝐾
75, 6cnextfval 22598 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → ((𝐽CnExt𝐾)‘𝐹) = 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
87releqd 5646 . 2 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → (Rel ((𝐽CnExt𝐾)‘𝐹) ↔ Rel 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))))
94, 8mpbiri 259 1 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → Rel ((𝐽CnExt𝐾)‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  wss 3933  {csn 4557   cuni 4830   ciun 4910   × cxp 5546  Rel wrel 5553  wf 6344  cfv 6348  (class class class)co 7145  t crest 16682  Topctop 21429  clsccl 21554  neicnei 21633   fLimf cflf 22471  CnExtccnext 22595
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-rep 5181  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-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-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  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-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-pm 8398  df-cnext 22596
This theorem is referenced by:  cnextfun  22600
  Copyright terms: Public domain W3C validator