Theorem estrreslem2 16778

Description: Lemma 2 for estrres 16779. (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

Step	Hyp	Ref	Expression
1		eqidd 2623	. . . 4 |- ( ph -> ( Base ` ndx ) = ( Base ` ndx ) )
2	1	3mix1d 1236	. . 3 |- ( ph -> ( ( Base ` ndx ) = ( Base ` ndx ) \/ ( Base ` ndx ) = ( Hom ` ndx ) \/ ( Base ` ndx ) = (comp ` ndx ) ) )
3		fvex 6201	. . . 4 |- ( Base ` ndx ) e. _V
4		eltpg 4227	. . . 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 ) ) ) )
5	3, 4	mp1i 13	. . 3 |- ( ph -> ( ( Base ` ndx ) e. { ( Base ` ndx ) , ( Hom ` ndx ) , (comp ` ndx ) } <-> ( ( Base ` ndx ) = ( Base ` ndx ) \/ ( Base ` ndx ) = ( Hom ` ndx ) \/ ( Base ` ndx ) = (comp ` ndx ) ) ) )
6	2, 5	mpbird 247	. 2 |- ( ph -> ( Base ` ndx ) e. { ( Base ` ndx ) , ( Hom ` ndx ) , (comp ` ndx ) } )
7		df-tp 4182	. . . . . 6 |- { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } = ( { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. } u. { <. (comp ` ndx ) , .x. >. } )
8	7	a1i 11	. . . . 5 |- ( ph -> { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } = ( { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. } u. { <. (comp ` ndx ) , .x. >. } ) )
9	8	dmeqd 5326	. . . 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 5331	. . . . 5 |- dom ( { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. } u. { <. (comp ` ndx ) , .x. >. } ) = ( dom { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. } u. dom { <. (comp ` ndx ) , .x. >. } )
11	10	a1i 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 5608	. . . . . 6 |- ( ( B e. V /\ H e. X ) -> dom { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. } = { ( Base ` ndx ) , ( Hom ` ndx ) } )
15	12, 13, 14	syl2anc 693	. . . . 5 |- ( ph -> dom { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. } = { ( Base ` ndx ) , ( Hom ` ndx ) } )
16		estrres.x	. . . . . 6 |- ( ph -> .x. e. Y )
17		dmsnopg 5606	. . . . . 6 |- ( .x. e. Y -> dom { <. (comp ` ndx ) , .x. >. } = { (comp ` ndx ) } )
18	16, 17	syl 17	. . . . 5 |- ( ph -> dom { <. (comp ` ndx ) , .x. >. } = { (comp ` ndx ) } )
19	15, 18	uneq12d 3768	. . . 4 |- ( ph -> ( dom { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. } u. dom { <. (comp ` ndx ) , .x. >. } ) = ( { ( Base ` ndx ) , ( Hom ` ndx ) } u. { (comp ` ndx ) } ) )
20	9, 11, 19	3eqtrd 2660	. . 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. >. } )
22	21	dmeqd 5326	. . 3 |- ( ph -> dom C = dom { <. ( Base ` ndx ) , B >. , <. ( Hom ` ndx ) , H >. , <. (comp ` ndx ) , .x. >. } )
23		df-tp 4182	. . . 4 |- { ( Base ` ndx ) , ( Hom ` ndx ) , (comp ` ndx ) } = ( { ( Base ` ndx ) , ( Hom ` ndx ) } u. { (comp ` ndx ) } )
24	23	a1i 11	. . 3 |- ( ph -> { ( Base ` ndx ) , ( Hom ` ndx ) , (comp ` ndx ) } = ( { ( Base ` ndx ) , ( Hom ` ndx ) } u. { (comp ` ndx ) } ) )
25	20, 22, 24	3eqtr4d 2666	. 2 |- ( ph -> dom C = { ( Base ` ndx ) , ( Hom ` ndx ) , (comp ` ndx ) } )
26	6, 25	eleqtrrd 2704	1 |- ( ph -> ( Base ` ndx ) e. dom C )

Syntax hints:   -> wi 4   <-> wb 196   \/ w3o 1036   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   {csn 4177   {cpr 4179   {ctp 4181   <.cop 4183   dom cdm 5114   ` cfv 5888   ndxcnx 15854   Basecbs 15857   Hom chom 15952  compcco 15953

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-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-dm 5124  df-iota 5851  df-fv 5896

This theorem is referenced by:  estrres  16779