Theorem trcleq2 13927

Description: Equality theorem for the transitive closure.

Assertion

Ref	Expression
trcleq2	|- (A = B -> Trcl(R, A, X) = Trcl(R, B, X))

Proof of Theorem trcleq2

Step	Hyp	Ref	Expression
1		predeq2 13882	. . . . . . . . . . 11 |- (A = B -> Pred(R, A, y) = Pred(R, B, y))
2	1	adantr 425	. . . . . . . . . 10 |- ((A = B /\ y e. a) -> Pred(R, A, y) = Pred(R, B, y))
3	2	iuneq2dv 3279	. . . . . . . . 9 |- (A = B -> U_y e. a Pred(R, A, y) = U_y e. a Pred(R, B, y))
4	3	eqeq2d 1895	. . . . . . . 8 |- (A = B -> (b = U_y e. a Pred(R, A, y) <-> b = U_y e. a Pred(R, B, y)))
5	4	opabbidv 3401	. . . . . . 7 |- (A = B -> {<.a, b>. | b = U_y e. a Pred(R, A, y)} = {<.a, b>. | b = U_y e. a Pred(R, B, y)})
6		rdgeq1 5142	. . . . . . 7 |- ({<.a, b>. | b = U_y e. a Pred(R, A, y)} = {<.a, b>. | b = U_y e. a Pred(R, B, y)} -> rec({<.a, b>. | b = U_y e. a Pred(R, A, y)}, Pred(R, A, X)) = rec({<.a, b>. | b = U_y e. a Pred(R, B, y)}, Pred(R, A, X)))
7	5, 6	syl 12	. . . . . 6 |- (A = B -> rec({<.a, b>. | b = U_y e. a Pred(R, A, y)}, Pred(R, A, X)) = rec({<.a, b>. | b = U_y e. a Pred(R, B, y)}, Pred(R, A, X)))
8		predeq2 13882	. . . . . . 7 |- (A = B -> Pred(R, A, X) = Pred(R, B, X))
9		rdgeq2 5143	. . . . . . 7 |- (Pred(R, A, X) = Pred(R, B, X) -> rec({<.a, b>. | b = U_y e. a Pred(R, B, y)}, Pred(R, A, X)) = rec({<.a, b>. | b = U_y e. a Pred(R, B, y)}, Pred(R, B, X)))
10	8, 9	syl 12	. . . . . 6 |- (A = B -> rec({<.a, b>. | b = U_y e. a Pred(R, B, y)}, Pred(R, A, X)) = rec({<.a, b>. | b = U_y e. a Pred(R, B, y)}, Pred(R, B, X)))
11	7, 10	eqtrd 1925	. . . . 5 |- (A = B -> rec({<.a, b>. | b = U_y e. a Pred(R, A, y)}, Pred(R, A, X)) = rec({<.a, b>. | b = U_y e. a Pred(R, B, y)}, Pred(R, B, X)))
12		reseq1 4218	. . . . 5 |- (rec({<.a, b>. | b = U_y e. a Pred(R, A, y)}, Pred(R, A, X)) = rec({<.a, b>. | b = U_y e. a Pred(R, B, y)}, Pred(R, B, X)) -> (rec({<.a, b>. | b = U_y e. a Pred(R, A, y)}, Pred(R, A, X)) |` om) = (rec({<.a, b>. | b = U_y e. a Pred(R, B, y)}, Pred(R, B, X)) |` om))
13	11, 12	syl 12	. . . 4 |- (A = B -> (rec({<.a, b>. | b = U_y e. a Pred(R, A, y)}, Pred(R, A, X)) |` om) = (rec({<.a, b>. | b = U_y e. a Pred(R, B, y)}, Pred(R, B, X)) |` om))
14	13	rneqd 4188	. . 3 |- (A = B -> ran (rec({<.a, b>. | b = U_y e. a Pred(R, A, y)}, Pred(R, A, X)) |` om) = ran (rec({<.a, b>. | b = U_y e. a Pred(R, B, y)}, Pred(R, B, X)) |` om))
15	14	unieqd 3188	. 2 |- (A = B -> U.ran (rec({<.a, b>. | b = U_y e. a Pred(R, A, y)}, Pred(R, A, X)) |` om) = U.ran (rec({<.a, b>. | b = U_y e. a Pred(R, B, y)}, Pred(R, B, X)) |` om))
16		df-trcl 13925	. 2 |- Trcl(R, A, X) = U.ran (rec({<.a, b>. | b = U_y e. a Pred(R, A, y)}, Pred(R, A, X)) |` om)
17		df-trcl 13925	. 2 |- Trcl(R, B, X) = U.ran (rec({<.a, b>. | b = U_y e. a Pred(R, B, y)}, Pred(R, B, X)) |` om)
18	15, 16, 17	3eqtr4g 1953	1 |- (A = B -> Trcl(R, A, X) = Trcl(R, B, X))

Syntax hints:   -> wi 3   = wceq 1298  U.cuni 3177  U_ciun 3255  {copab 3395  omcom 3949  ran crn 3987   |` cres 3988  reccrdg 5139  Predcpred 13879  Trclctrcl 13924

This theorem is referenced by:  trcleq2d 13930

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-iun 3257  df-br 3339  df-opab 3396  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-rdg 5140  df-pred 13880  df-trcl 13925

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