Theorem cnextrel 21867

Description: In the general case, a continuous extension is a relation. (Contributed by Thierry Arnoux, 20-Dec-2017.)

Hypotheses

Ref	Expression
cnextfrel.1	|- C = U. J
cnextfrel.2	|- B = U. K

Assertion

Ref	Expression
cnextrel	|- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ C ) ) -> Rel ( ( JCnExt K ) ` F ) )

Proof of Theorem cnextrel

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		relxp 5227	. . . 4 |- Rel ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) )
2	1	rgenw 2924	. . 3 |- A. x e. ( ( cls ` J ) ` A ) Rel ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) )
3		reliun 5239	. . 3 |- ( Rel U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) <-> A. x e. ( ( cls ` J ) ` A ) Rel ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) )
4	2, 3	mpbir 221	. 2 |- Rel U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) )
5		cnextfrel.1	. . . 4 |- C = U. J
6		cnextfrel.2	. . . 4 |- B = U. K
7	5, 6	cnextfval 21866	. . 3 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ C ) ) -> ( ( JCnExt K ) ` F ) = U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) )
8	7	releqd 5203	. 2 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ C ) ) -> ( Rel ( ( JCnExt K ) ` F ) <-> Rel U_ x e. ( ( cls ` J ) ` A ) ( { x } X. ( ( K fLimf ( ( ( nei ` J ) ` { x } ) ↾t A ) ) ` F ) ) ) )
9	4, 8	mpbiri 248	1 |- ( ( ( J e. Top /\ K e. Top ) /\ ( F : A --> B /\ A C_ C ) ) -> Rel ( ( JCnExt K ) ` F ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   {csn 4177   U.cuni 4436   U_ciun 4520   X. cxp 5112   Rel wrel 5119   -->wf 5884   ` cfv 5888  (class class class)co 6650   ↾_t crest 16081   Topctop 20698   clsccl 20822   neicnei 20901   fLimf cflf 21739  CnExtccnext 21863

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pm 7860  df-cnext 21864

This theorem is referenced by:  cnextfun  21868