Theorem cstucnd 22088

Description: A constant function is uniformly continuous. Deduction form. Example 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 16-Nov-2017.)

Hypotheses

Ref	Expression
cstucnd.1	|- ( ph -> U e. (UnifOn ` X ) )
cstucnd.2	|- ( ph -> V e. (UnifOn ` Y ) )
cstucnd.3	|- ( ph -> A e. Y )

Assertion

Ref	Expression
cstucnd	|- ( ph -> ( X X. { A } ) e. ( U Cnu V ) )

Proof of Theorem cstucnd

Dummy variables s r x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cstucnd.3	. . 3 |- ( ph -> A e. Y )
2		fconst6g 6094	. . 3 |- ( A e. Y -> ( X X. { A } ) : X --> Y )
3	1, 2	syl 17	. 2 |- ( ph -> ( X X. { A } ) : X --> Y )
4		cstucnd.1	. . . . . 6 |- ( ph -> U e. (UnifOn ` X ) )
5	4	adantr 481	. . . . 5 |- ( ( ph /\ s e. V ) -> U e. (UnifOn ` X ) )
6		ustne0 22017	. . . . 5 |- ( U e. (UnifOn ` X ) -> U =/= (/) )
7	5, 6	syl 17	. . . 4 |- ( ( ph /\ s e. V ) -> U =/= (/) )
8		cstucnd.2	. . . . . . . . . 10 |- ( ph -> V e. (UnifOn ` Y ) )
9	8	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ s e. V ) /\ r e. U ) /\ ( x e. X /\ y e. X ) ) -> V e. (UnifOn ` Y ) )
10		simpllr 799	. . . . . . . . 9 |- ( ( ( ( ph /\ s e. V ) /\ r e. U ) /\ ( x e. X /\ y e. X ) ) -> s e. V )
11	1	ad3antrrr 766	. . . . . . . . 9 |- ( ( ( ( ph /\ s e. V ) /\ r e. U ) /\ ( x e. X /\ y e. X ) ) -> A e. Y )
12		ustref 22022	. . . . . . . . 9 |- ( ( V e. (UnifOn ` Y ) /\ s e. V /\ A e. Y ) -> A s A )
13	9, 10, 11, 12	syl3anc 1326	. . . . . . . 8 |- ( ( ( ( ph /\ s e. V ) /\ r e. U ) /\ ( x e. X /\ y e. X ) ) -> A s A )
14		simprl 794	. . . . . . . . 9 |- ( ( ( ( ph /\ s e. V ) /\ r e. U ) /\ ( x e. X /\ y e. X ) ) -> x e. X )
15		fvconst2g 6467	. . . . . . . . 9 |- ( ( A e. Y /\ x e. X ) -> ( ( X X. { A } ) ` x ) = A )
16	11, 14, 15	syl2anc 693	. . . . . . . 8 |- ( ( ( ( ph /\ s e. V ) /\ r e. U ) /\ ( x e. X /\ y e. X ) ) -> ( ( X X. { A } ) ` x ) = A )
17		simprr 796	. . . . . . . . 9 |- ( ( ( ( ph /\ s e. V ) /\ r e. U ) /\ ( x e. X /\ y e. X ) ) -> y e. X )
18		fvconst2g 6467	. . . . . . . . 9 |- ( ( A e. Y /\ y e. X ) -> ( ( X X. { A } ) ` y ) = A )
19	11, 17, 18	syl2anc 693	. . . . . . . 8 |- ( ( ( ( ph /\ s e. V ) /\ r e. U ) /\ ( x e. X /\ y e. X ) ) -> ( ( X X. { A } ) ` y ) = A )
20	13, 16, 19	3brtr4d 4685	. . . . . . 7 |- ( ( ( ( ph /\ s e. V ) /\ r e. U ) /\ ( x e. X /\ y e. X ) ) -> ( ( X X. { A } ) ` x ) s ( ( X X. { A } ) ` y ) )
21	20	a1d 25	. . . . . 6 |- ( ( ( ( ph /\ s e. V ) /\ r e. U ) /\ ( x e. X /\ y e. X ) ) -> ( x r y -> ( ( X X. { A } ) ` x ) s ( ( X X. { A } ) ` y ) ) )
22	21	ralrimivva 2971	. . . . 5 |- ( ( ( ph /\ s e. V ) /\ r e. U ) -> A. x e. X A. y e. X ( x r y -> ( ( X X. { A } ) ` x ) s ( ( X X. { A } ) ` y ) ) )
23	22	reximdva0 3933	. . . 4 |- ( ( ( ph /\ s e. V ) /\ U =/= (/) ) -> E. r e. U A. x e. X A. y e. X ( x r y -> ( ( X X. { A } ) ` x ) s ( ( X X. { A } ) ` y ) ) )
24	7, 23	mpdan 702	. . 3 |- ( ( ph /\ s e. V ) -> E. r e. U A. x e. X A. y e. X ( x r y -> ( ( X X. { A } ) ` x ) s ( ( X X. { A } ) ` y ) ) )
25	24	ralrimiva 2966	. 2 |- ( ph -> A. s e. V E. r e. U A. x e. X A. y e. X ( x r y -> ( ( X X. { A } ) ` x ) s ( ( X X. { A } ) ` y ) ) )
26		isucn 22082	. . 3 |- ( ( U e. (UnifOn ` X ) /\ V e. (UnifOn ` Y ) ) -> ( ( X X. { A } ) e. ( U Cnu V ) <-> ( ( X X. { A } ) : X --> Y /\ A. s e. V E. r e. U A. x e. X A. y e. X ( x r y -> ( ( X X. { A } ) ` x ) s ( ( X X. { A } ) ` y ) ) ) ) )
27	4, 8, 26	syl2anc 693	. 2 |- ( ph -> ( ( X X. { A } ) e. ( U Cnu V ) <-> ( ( X X. { A } ) : X --> Y /\ A. s e. V E. r e. U A. x e. X A. y e. X ( x r y -> ( ( X X. { A } ) ` x ) s ( ( X X. { A } ) ` y ) ) ) ) )
28	3, 25, 27	mpbir2and 957	1 |- ( ph -> ( X X. { A } ) e. ( U Cnu V ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   (/)c0 3915   {csn 4177   class class class wbr 4653   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650  UnifOncust 22003   Cn_ucucn 22079

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-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-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-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-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-ust 22004  df-ucn 22080

This theorem is referenced by: (None)