Theorem ucnprima 22086

Description: The preimage by a uniformly continuous function F of an entourage W of Y is an entourage of X. Note of the definition 1 of [BourbakiTop1] p. II.6. (Contributed by Thierry Arnoux, 19-Nov-2017.)

Hypotheses

Ref	Expression
ucnprima.1	|- ( ph -> U e. (UnifOn ` X ) )
ucnprima.2	|- ( ph -> V e. (UnifOn ` Y ) )
ucnprima.3	|- ( ph -> F e. ( U Cnu V ) )
ucnprima.4	|- ( ph -> W e. V )
ucnprima.5	|- G = ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. )

Assertion

Ref	Expression
ucnprima	|- ( ph -> ( `' G " W ) e. U )

Distinct variable groups:   x, y, F   x, X, y   x, G, y   x, U, y   x, V   x, W, y   x, Y   ph, x, y

Allowed substitution hints:   V( y)   Y( y)

Proof of Theorem ucnprima

Dummy variable r is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ucnprima.1	. . . 4 |- ( ph -> U e. (UnifOn ` X ) )
2		ucnprima.2	. . . 4 |- ( ph -> V e. (UnifOn ` Y ) )
3		ucnprima.3	. . . 4 |- ( ph -> F e. ( U Cnu V ) )
4		ucnprima.4	. . . 4 |- ( ph -> W e. V )
5		ucnprima.5	. . . 4 |- G = ( x e. X , y e. X |-> <. ( F ` x ) , ( F ` y ) >. )
6	1, 2, 3, 4, 5	ucnima 22085	. . 3 |- ( ph -> E. r e. U ( G " r ) C_ W )
7	5	mpt2fun 6762	. . . . 5 |- Fun G
8		ustssxp 22008	. . . . . . 7 |- ( ( U e. (UnifOn ` X ) /\ r e. U ) -> r C_ ( X X. X ) )
9	1, 8	sylan 488	. . . . . 6 |- ( ( ph /\ r e. U ) -> r C_ ( X X. X ) )
10		opex 4932	. . . . . . 7 |- <. ( F ` x ) , ( F ` y ) >. e. _V
11	5, 10	dmmpt2 7240	. . . . . 6 |- dom G = ( X X. X )
12	9, 11	syl6sseqr 3652	. . . . 5 |- ( ( ph /\ r e. U ) -> r C_ dom G )
13		funimass3 6333	. . . . 5 |- ( ( Fun G /\ r C_ dom G ) -> ( ( G " r ) C_ W <-> r C_ ( `' G " W ) ) )
14	7, 12, 13	sylancr 695	. . . 4 |- ( ( ph /\ r e. U ) -> ( ( G " r ) C_ W <-> r C_ ( `' G " W ) ) )
15	14	rexbidva 3049	. . 3 |- ( ph -> ( E. r e. U ( G " r ) C_ W <-> E. r e. U r C_ ( `' G " W ) ) )
16	6, 15	mpbid 222	. 2 |- ( ph -> E. r e. U r C_ ( `' G " W ) )
17	1	adantr 481	. . . 4 |- ( ( ph /\ r e. U ) -> U e. (UnifOn ` X ) )
18		simpr 477	. . . 4 |- ( ( ph /\ r e. U ) -> r e. U )
19		cnvimass 5485	. . . . . 6 |- ( `' G " W ) C_ dom G
20	19, 11	sseqtri 3637	. . . . 5 |- ( `' G " W ) C_ ( X X. X )
21	20	a1i 11	. . . 4 |- ( ( ph /\ r e. U ) -> ( `' G " W ) C_ ( X X. X ) )
22		ustssel 22009	. . . 4 |- ( ( U e. (UnifOn ` X ) /\ r e. U /\ ( `' G " W ) C_ ( X X. X ) ) -> ( r C_ ( `' G " W ) -> ( `' G " W ) e. U ) )
23	17, 18, 21, 22	syl3anc 1326	. . 3 |- ( ( ph /\ r e. U ) -> ( r C_ ( `' G " W ) -> ( `' G " W ) e. U ) )
24	23	rexlimdva 3031	. 2 |- ( ph -> ( E. r e. U r C_ ( `' G " W ) -> ( `' G " W ) e. U ) )
25	16, 24	mpd 15	1 |- ( ph -> ( `' G " W ) e. U )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   E.wrex 2913   C_ wss 3574   <.cop 4183   X. cxp 5112   `'ccnv 5113   dom cdm 5114   "cima 5117   Fun wfun 5882   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652  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-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-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-ust 22004  df-ucn 22080

This theorem is referenced by:  fmucnd  22096