Theorem reftr 15497

Description: Refinement is transitive.

Assertion

Ref	Expression
reftr	|- ((C e. D /\ ARefB /\ BRefC) -> ARefC)

Proof of Theorem reftr

Step	Hyp	Ref	Expression
1		eqid 1884	. . . 4 |- U.A = U.A
2		eqid 1884	. . . 4 |- U.C = U.C
3	1, 2	isref 15488	. . 3 |- (C e. D -> (ARefC <-> (U.A = U.C /\ A.x e. C E.z e. A x C_ z)))
4	3	3ad2ant1 897	. 2 |- ((C e. D /\ ARefB /\ BRefC) -> (ARefC <-> (U.A = U.C /\ A.x e. C E.z e. A x C_ z)))
5		refrel 15487	. . . . . 6 |- Rel Ref
6	5	brrelexi 4029	. . . . 5 |- (BRefC -> B e. _V)
7	6	3ad2ant3 899	. . . 4 |- ((C e. D /\ ARefB /\ BRefC) -> B e. _V)
8		simp2 877	. . . 4 |- ((C e. D /\ ARefB /\ BRefC) -> ARefB)
9		eqid 1884	. . . . 5 |- U.B = U.B
10	1, 9	refbas 15489	. . . 4 |- ((B e. _V /\ ARefB) -> U.A = U.B)
11	7, 8, 10	syl11anc 524	. . 3 |- ((C e. D /\ ARefB /\ BRefC) -> U.A = U.B)
12	9, 2	refbas 15489	. . . 4 |- ((C e. D /\ BRefC) -> U.B = U.C)
13	12	3adant2 895	. . 3 |- ((C e. D /\ ARefB /\ BRefC) -> U.B = U.C)
14	11, 13	eqtrd 1925	. 2 |- ((C e. D /\ ARefB /\ BRefC) -> U.A = U.C)
15		refssex 15490	. . . . . 6 |- ((C e. D /\ BRefC /\ x e. C) -> E.y e. B x C_ y)
16	15	3expia 1069	. . . . 5 |- ((C e. D /\ BRefC) -> (x e. C -> E.y e. B x C_ y))
17	16	3adant2 895	. . . 4 |- ((C e. D /\ ARefB /\ BRefC) -> (x e. C -> E.y e. B x C_ y))
18	7	adantr 425	. . . . . . . 8 |- (((C e. D /\ ARefB /\ BRefC) /\ (y e. B /\ x C_ y)) -> B e. _V)
19		simpl2 880	. . . . . . . 8 |- (((C e. D /\ ARefB /\ BRefC) /\ (y e. B /\ x C_ y)) -> ARefB)
20		simprl 450	. . . . . . . 8 |- (((C e. D /\ ARefB /\ BRefC) /\ (y e. B /\ x C_ y)) -> y e. B)
21		refssex 15490	. . . . . . . 8 |- ((B e. _V /\ ARefB /\ y e. B) -> E.z e. A y C_ z)
22	18, 19, 20, 21	syl111anc 1100	. . . . . . 7 |- (((C e. D /\ ARefB /\ BRefC) /\ (y e. B /\ x C_ y)) -> E.z e. A y C_ z)
23		sstr2 2623	. . . . . . . . 9 |- (x C_ y -> (y C_ z -> x C_ z))
24	23	reximdv 2202	. . . . . . . 8 |- (x C_ y -> (E.z e. A y C_ z -> E.z e. A x C_ z))
25	24	ad2antll 443	. . . . . . 7 |- (((C e. D /\ ARefB /\ BRefC) /\ (y e. B /\ x C_ y)) -> (E.z e. A y C_ z -> E.z e. A x C_ z))
26	22, 25	mpd 29	. . . . . 6 |- (((C e. D /\ ARefB /\ BRefC) /\ (y e. B /\ x C_ y)) -> E.z e. A x C_ z)
27	26	exp32 408	. . . . 5 |- ((C e. D /\ ARefB /\ BRefC) -> (y e. B -> (x C_ y -> E.z e. A x C_ z)))
28	27	r19.23adv 2215	. . . 4 |- ((C e. D /\ ARefB /\ BRefC) -> (E.y e. B x C_ y -> E.z e. A x C_ z))
29	17, 28	syld 30	. . 3 |- ((C e. D /\ ARefB /\ BRefC) -> (x e. C -> E.z e. A x C_ z))
30	29	r19.21aiv 2175	. 2 |- ((C e. D /\ ARefB /\ BRefC) -> A.x e. C E.z e. A x C_ z)
31	4, 14, 30	mpbir2and 802	1 |- ((C e. D /\ ARefB /\ BRefC) -> ARefC)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106  _Vcvv 2292   C_ wss 2593  U.cuni 3177   class class class wbr 3338  Refcref 15458

This theorem is referenced by:  refssfne 15504

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-rel 4001  df-ref 15464

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