Theorem setsabsd 13184

Description: Replacing the same components twice yields the same as the second setting only. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Jim Kingdon, 22-Jan-2023.)

Hypotheses

Ref	Expression
setsabsd.s	|- ( ph -> S e. V )
setsabsd.a	|- ( ph -> A e. W )
setsabsd.b	|- ( ph -> B e. X )
setsabsd.c	|- ( ph -> C e. U )

Assertion

Ref	Expression
setsabsd	|- ( ph -> ( ( S sSet <. A , B >. ) sSet <. A , C >. ) = ( S sSet <. A , C >. ) )

Proof of Theorem setsabsd

Step	Hyp	Ref	Expression
1		setsabsd.s	. . . 4 |- ( ph -> S e. V )
2		setsabsd.a	. . . 4 |- ( ph -> A e. W )
3		setsabsd.b	. . . 4 |- ( ph -> B e. X )
4		setsresg 13183	. . . 4 |- ( ( S e. V /\ A e. W /\ B e. X ) -> ( ( S sSet <. A , B >. ) |` ( _V \ { A } ) ) = ( S |` ( _V \ { A } ) ) )
5	1, 2, 3, 4	syl3anc 1274	. . 3 |- ( ph -> ( ( S sSet <. A , B >. ) |` ( _V \ { A } ) ) = ( S |` ( _V \ { A } ) ) )
6	5	uneq1d 3362	. 2 |- ( ph -> ( ( ( S sSet <. A , B >. ) |` ( _V \ { A } ) ) u. { <. A , C >. } ) = ( ( S |` ( _V \ { A } ) ) u. { <. A , C >. } ) )
7		setsex 13177	. . . 4 |- ( ( S e. V /\ A e. W /\ B e. X ) -> ( S sSet <. A , B >. ) e. _V )
8	1, 2, 3, 7	syl3anc 1274	. . 3 |- ( ph -> ( S sSet <. A , B >. ) e. _V )
9		setsabsd.c	. . 3 |- ( ph -> C e. U )
10		setsvala 13176	. . 3 |- ( ( ( S sSet <. A , B >. ) e. _V /\ A e. W /\ C e. U ) -> ( ( S sSet <. A , B >. ) sSet <. A , C >. ) = ( ( ( S sSet <. A , B >. ) |` ( _V \ { A } ) ) u. { <. A , C >. } ) )
11	8, 2, 9, 10	syl3anc 1274	. 2 |- ( ph -> ( ( S sSet <. A , B >. ) sSet <. A , C >. ) = ( ( ( S sSet <. A , B >. ) |` ( _V \ { A } ) ) u. { <. A , C >. } ) )
12		setsvala 13176	. . 3 |- ( ( S e. V /\ A e. W /\ C e. U ) -> ( S sSet <. A , C >. ) = ( ( S |` ( _V \ { A } ) ) u. { <. A , C >. } ) )
13	1, 2, 9, 12	syl3anc 1274	. 2 |- ( ph -> ( S sSet <. A , C >. ) = ( ( S |` ( _V \ { A } ) ) u. { <. A , C >. } ) )
14	6, 11, 13	3eqtr4d 2274	1 |- ( ph -> ( ( S sSet <. A , B >. ) sSet <. A , C >. ) = ( S sSet <. A , C >. ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2202   _Vcvv 2803   \ cdif 3198   u. cun 3199   {csn 3673   <.cop 3676   |` cres 4733  (class class class)co 6028   sSet csts 13143

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-res 4743  df-iota 5293  df-fun 5335  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-sets 13152

This theorem is referenced by:  ressressg  13221