Theorem basprssdmsets 15925

Description: The pair of the base index and another index is a subset of the domain of the structure obtained by replacing/adding a slot at the other index in a structure having a base slot. (Contributed by AV, 7-Jun-2021.) (Revised by AV, 16-Nov-2021.)

Hypotheses

Ref	Expression
basprssdmsets.s	|- ( ph -> S Struct X )
basprssdmsets.i	|- ( ph -> I e. U )
basprssdmsets.w	|- ( ph -> E e. W )
basprssdmsets.b	|- ( ph -> ( Base ` ndx ) e. dom S )

Assertion

Ref	Expression
basprssdmsets	|- ( ph -> { ( Base ` ndx ) , I } C_ dom ( S sSet <. I , E >. ) )

Proof of Theorem basprssdmsets

Step	Hyp	Ref	Expression
1		basprssdmsets.b	. . . . 5 |- ( ph -> ( Base ` ndx ) e. dom S )
2	1	orcd 407	. . . 4 |- ( ph -> ( ( Base ` ndx ) e. dom S \/ ( Base ` ndx ) e. { I } ) )
3		elun 3753	. . . 4 |- ( ( Base ` ndx ) e. ( dom S u. { I } ) <-> ( ( Base ` ndx ) e. dom S \/ ( Base ` ndx ) e. { I } ) )
4	2, 3	sylibr 224	. . 3 |- ( ph -> ( Base ` ndx ) e. ( dom S u. { I } ) )
5		basprssdmsets.i	. . . . . 6 |- ( ph -> I e. U )
6		snidg 4206	. . . . . 6 |- ( I e. U -> I e. { I } )
7	5, 6	syl 17	. . . . 5 |- ( ph -> I e. { I } )
8	7	olcd 408	. . . 4 |- ( ph -> ( I e. dom S \/ I e. { I } ) )
9		elun 3753	. . . 4 |- ( I e. ( dom S u. { I } ) <-> ( I e. dom S \/ I e. { I } ) )
10	8, 9	sylibr 224	. . 3 |- ( ph -> I e. ( dom S u. { I } ) )
11	4, 10	prssd 4354	. 2 |- ( ph -> { ( Base ` ndx ) , I } C_ ( dom S u. { I } ) )
12		basprssdmsets.s	. . . 4 |- ( ph -> S Struct X )
13		structex 15868	. . . 4 |- ( S Struct X -> S e. _V )
14	12, 13	syl 17	. . 3 |- ( ph -> S e. _V )
15		basprssdmsets.w	. . 3 |- ( ph -> E e. W )
16		setsdm 15892	. . 3 |- ( ( S e. _V /\ E e. W ) -> dom ( S sSet <. I , E >. ) = ( dom S u. { I } ) )
17	14, 15, 16	syl2anc 693	. 2 |- ( ph -> dom ( S sSet <. I , E >. ) = ( dom S u. { I } ) )
18	11, 17	sseqtr4d 3642	1 |- ( ph -> { ( Base ` ndx ) , I } C_ dom ( S sSet <. I , E >. ) )

Syntax hints:   -> wi 4   \/ wo 383   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   C_ wss 3574   {csn 4177   {cpr 4179   <.cop 4183   class class class wbr 4653   dom cdm 5114   ` cfv 5888  (class class class)co 6650   Struct cstr 15853   ndxcnx 15854   sSet csts 15855   Basecbs 15857

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-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-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-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-res 5126  df-iota 5851  df-fun 5890  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-struct 15859  df-sets 15864

This theorem is referenced by:  setsvtx  25927  setsiedg  25928