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Theorem estrreslem2 16101
Description: Lemma 2 for estrres 16102. (Contributed by AV, 14-Mar-2020.)
Hypotheses
Ref Expression
estrres.c  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
estrres.b  |-  ( ph  ->  B  e.  V )
estrres.h  |-  ( ph  ->  H  e.  X )
estrres.x  |-  ( ph  ->  .x.  e.  Y )
Assertion
Ref Expression
estrreslem2  |-  ( ph  ->  ( Base `  ndx )  e.  dom  C )

Proof of Theorem estrreslem2
StepHypRef Expression
1 eqidd 2472 . . . 4  |-  ( ph  ->  ( Base `  ndx )  =  ( Base ` 
ndx ) )
213mix1d 1205 . . 3  |-  ( ph  ->  ( ( Base `  ndx )  =  ( Base ` 
ndx )  \/  ( Base `  ndx )  =  ( Hom  `  ndx )  \/  ( Base ` 
ndx )  =  (comp `  ndx ) ) )
3 fvex 5889 . . . 4  |-  ( Base `  ndx )  e.  _V
4 eltpg 4005 . . . 4  |-  ( (
Base `  ndx )  e. 
_V  ->  ( ( Base `  ndx )  e.  {
( Base `  ndx ) ,  ( Hom  `  ndx ) ,  (comp `  ndx ) }  <->  ( ( Base `  ndx )  =  (
Base `  ndx )  \/  ( Base `  ndx )  =  ( Hom  ` 
ndx )  \/  ( Base `  ndx )  =  (comp `  ndx ) ) ) )
53, 4mp1i 13 . . 3  |-  ( ph  ->  ( ( Base `  ndx )  e.  { ( Base `  ndx ) ,  ( Hom  `  ndx ) ,  (comp `  ndx ) }  <->  ( ( Base `  ndx )  =  (
Base `  ndx )  \/  ( Base `  ndx )  =  ( Hom  ` 
ndx )  \/  ( Base `  ndx )  =  (comp `  ndx ) ) ) )
62, 5mpbird 240 . 2  |-  ( ph  ->  ( Base `  ndx )  e.  { ( Base `  ndx ) ,  ( Hom  `  ndx ) ,  (comp `  ndx ) } )
7 df-tp 3964 . . . . . 6  |-  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. }  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  H >. }  u.  { <. (comp ` 
ndx ) ,  .x.  >. } )
87a1i 11 . . . . 5  |-  ( ph  ->  { <. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. }  =  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  H >. }  u.  { <. (comp ` 
ndx ) ,  .x.  >. } ) )
98dmeqd 5042 . . . 4  |-  ( ph  ->  dom  { <. ( Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. }  =  dom  ( {
<. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  H >. }  u.  { <. (comp ` 
ndx ) ,  .x.  >. } ) )
10 dmun 5047 . . . . 5  |-  dom  ( { <. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  H >. }  u.  { <. (comp ` 
ndx ) ,  .x.  >. } )  =  ( dom  { <. ( Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. }  u.  dom  {
<. (comp `  ndx ) , 
.x.  >. } )
1110a1i 11 . . . 4  |-  ( ph  ->  dom  ( { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. }  u.  { <. (comp `  ndx ) , 
.x.  >. } )  =  ( dom  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. }  u.  dom  {
<. (comp `  ndx ) , 
.x.  >. } ) )
12 estrres.b . . . . . 6  |-  ( ph  ->  B  e.  V )
13 estrres.h . . . . . 6  |-  ( ph  ->  H  e.  X )
14 dmpropg 5316 . . . . . 6  |-  ( ( B  e.  V  /\  H  e.  X )  ->  dom  { <. ( Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. }  =  {
( Base `  ndx ) ,  ( Hom  `  ndx ) } )
1512, 13, 14syl2anc 673 . . . . 5  |-  ( ph  ->  dom  { <. ( Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. }  =  {
( Base `  ndx ) ,  ( Hom  `  ndx ) } )
16 estrres.x . . . . . 6  |-  ( ph  ->  .x.  e.  Y )
17 dmsnopg 5314 . . . . . 6  |-  (  .x.  e.  Y  ->  dom  { <. (comp `  ndx ) , 
.x.  >. }  =  {
(comp `  ndx ) } )
1816, 17syl 17 . . . . 5  |-  ( ph  ->  dom  { <. (comp ` 
ndx ) ,  .x.  >. }  =  { (comp ` 
ndx ) } )
1915, 18uneq12d 3580 . . . 4  |-  ( ph  ->  ( dom  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. }  u.  dom  {
<. (comp `  ndx ) , 
.x.  >. } )  =  ( { ( Base `  ndx ) ,  ( Hom  `  ndx ) }  u.  { (comp `  ndx ) } ) )
209, 11, 193eqtrd 2509 . . 3  |-  ( ph  ->  dom  { <. ( Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. }  =  ( {
( Base `  ndx ) ,  ( Hom  `  ndx ) }  u.  { (comp `  ndx ) } ) )
21 estrres.c . . . 4  |-  ( ph  ->  C  =  { <. (
Base `  ndx ) ,  B >. ,  <. ( Hom  `  ndx ) ,  H >. ,  <. (comp ` 
ndx ) ,  .x.  >. } )
2221dmeqd 5042 . . 3  |-  ( ph  ->  dom  C  =  dom  {
<. ( Base `  ndx ) ,  B >. , 
<. ( Hom  `  ndx ) ,  H >. , 
<. (comp `  ndx ) , 
.x.  >. } )
23 df-tp 3964 . . . 4  |-  { (
Base `  ndx ) ,  ( Hom  `  ndx ) ,  (comp `  ndx ) }  =  ( { ( Base `  ndx ) ,  ( Hom  ` 
ndx ) }  u.  { (comp `  ndx ) } )
2423a1i 11 . . 3  |-  ( ph  ->  { ( Base `  ndx ) ,  ( Hom  ` 
ndx ) ,  (comp `  ndx ) }  =  ( { ( Base `  ndx ) ,  ( Hom  ` 
ndx ) }  u.  { (comp `  ndx ) } ) )
2520, 22, 243eqtr4d 2515 . 2  |-  ( ph  ->  dom  C  =  {
( Base `  ndx ) ,  ( Hom  `  ndx ) ,  (comp `  ndx ) } )
266, 25eleqtrrd 2552 1  |-  ( ph  ->  ( Base `  ndx )  e.  dom  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ w3o 1006    = wceq 1452    e. wcel 1904   _Vcvv 3031    u. cun 3388   {csn 3959   {cpr 3961   {ctp 3963   <.cop 3965   dom cdm 4839   ` cfv 5589   ndxcnx 15196   Basecbs 15199   Hom chom 15279  compcco 15280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-br 4396  df-dm 4849  df-iota 5553  df-fv 5597
This theorem is referenced by:  estrres  16102
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