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Theorem cnextrel 22671
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 5541 . . . 4 Rel ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
21rgenw 3121 . . 3 𝑥 ∈ ((cls‘𝐽)‘𝐴)Rel ({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))
3 reliun 5657 . . 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 234 . 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 22670 . . 3 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → ((𝐽CnExt𝐾)‘𝐹) = 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))
87releqd 5621 . 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 261 1 (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴𝐵𝐴𝐶)) → Rel ((𝐽CnExt𝐾)‘𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2112  wral 3109  wss 3884  {csn 4528   cuni 4803   ciun 4884   × cxp 5521  Rel wrel 5528  wf 6324  cfv 6328  (class class class)co 7139  t crest 16689  Topctop 21501  clsccl 21626  neicnei 21705   fLimf cflf 22543  CnExtccnext 22667
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-rep 5157  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-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-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  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-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-pm 8396  df-cnext 22668
This theorem is referenced by:  cnextfun  22672
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