Theorem reftr 21317

Description: Refinement is transitive. (Contributed by Jeff Hankins, 18-Jan-2010.) (Revised by Thierry Arnoux, 3-Feb-2020.)

Assertion

Ref	Expression
reftr	|- ( ( A Ref B /\ B Ref C ) -> A Ref C )

Proof of Theorem reftr

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2622	. . . 4 |- U. B = U. B
2		eqid 2622	. . . 4 |- U. C = U. C
3	1, 2	refbas 21313	. . 3 |- ( B Ref C -> U. C = U. B )
4		eqid 2622	. . . 4 |- U. A = U. A
5	4, 1	refbas 21313	. . 3 |- ( A Ref B -> U. B = U. A )
6	3, 5	sylan9eqr 2678	. 2 |- ( ( A Ref B /\ B Ref C ) -> U. C = U. A )
7		refssex 21314	. . . . . 6 |- ( ( A Ref B /\ x e. A ) -> E. y e. B x C_ y )
8	7	ex 450	. . . . 5 |- ( A Ref B -> ( x e. A -> E. y e. B x C_ y ) )
9	8	adantr 481	. . . 4 |- ( ( A Ref B /\ B Ref C ) -> ( x e. A -> E. y e. B x C_ y ) )
10		refssex 21314	. . . . . . 7 |- ( ( B Ref C /\ y e. B ) -> E. z e. C y C_ z )
11	10	ad2ant2lr 784	. . . . . 6 |- ( ( ( A Ref B /\ B Ref C ) /\ ( y e. B /\ x C_ y ) ) -> E. z e. C y C_ z )
12		sstr2 3610	. . . . . . . 8 |- ( x C_ y -> ( y C_ z -> x C_ z ) )
13	12	reximdv 3016	. . . . . . 7 |- ( x C_ y -> ( E. z e. C y C_ z -> E. z e. C x C_ z ) )
14	13	ad2antll 765	. . . . . 6 |- ( ( ( A Ref B /\ B Ref C ) /\ ( y e. B /\ x C_ y ) ) -> ( E. z e. C y C_ z -> E. z e. C x C_ z ) )
15	11, 14	mpd 15	. . . . 5 |- ( ( ( A Ref B /\ B Ref C ) /\ ( y e. B /\ x C_ y ) ) -> E. z e. C x C_ z )
16	15	rexlimdvaa 3032	. . . 4 |- ( ( A Ref B /\ B Ref C ) -> ( E. y e. B x C_ y -> E. z e. C x C_ z ) )
17	9, 16	syld 47	. . 3 |- ( ( A Ref B /\ B Ref C ) -> ( x e. A -> E. z e. C x C_ z ) )
18	17	ralrimiv 2965	. 2 |- ( ( A Ref B /\ B Ref C ) -> A. x e. A E. z e. C x C_ z )
19		refrel 21311	. . . . 5 |- Rel Ref
20	19	brrelexi 5158	. . . 4 |- ( A Ref B -> A e. _V )
21	20	adantr 481	. . 3 |- ( ( A Ref B /\ B Ref C ) -> A e. _V )
22	4, 2	isref 21312	. . 3 |- ( A e. _V -> ( A Ref C <-> ( U. C = U. A /\ A. x e. A E. z e. C x C_ z ) ) )
23	21, 22	syl 17	. 2 |- ( ( A Ref B /\ B Ref C ) -> ( A Ref C <-> ( U. C = U. A /\ A. x e. A E. z e. C x C_ z ) ) )
24	6, 18, 23	mpbir2and 957	1 |- ( ( A Ref B /\ B Ref C ) -> A Ref C )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   _Vcvv 3200   C_ wss 3574   U.cuni 4436   class class class wbr 4653   Refcref 21305

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-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-ref 21308

This theorem is referenced by:  refssfne  32353