Theorem riota5 5580

Description: A method for computing restricted iota.

Hypothesis

Ref	Expression
riota5.1	|- ((ph /\ B e. A /\ x e. A) -> (ps <-> x = B))

Assertion

Ref	Expression
riota5	|- ((ph /\ B e. A) -> (iota_x e. Aps) = B)

Distinct variable groups:   x,A   x,B   ph,x

Proof of Theorem riota5

Step	Hyp	Ref	Expression
1		elex 2302	. . . 4 |- (B e. A -> E.x x = B)
2	1	adantl 424	. . 3 |- ((ph /\ B e. A) -> E.x x = B)
3		simpr 350	. . . . . . . 8 |- (((ph /\ B e. A) /\ x = B) -> x = B)
4		simpll 448	. . . . . . . 8 |- (((ph /\ B e. A) /\ x = B) -> ph)
5		simplr 449	. . . . . . . 8 |- (((ph /\ B e. A) /\ x = B) -> B e. A)
6		eleq1a 1966	. . . . . . . . . 10 |- (B e. A -> (x = B -> x e. A))
7	6	imp 377	. . . . . . . . 9 |- ((B e. A /\ x = B) -> x e. A)
8	7	adantll 428	. . . . . . . 8 |- (((ph /\ B e. A) /\ x = B) -> x e. A)
9		riota5.1	. . . . . . . . 9 |- ((ph /\ B e. A /\ x e. A) -> (ps <-> x = B))
10	9	biimparc 463	. . . . . . . 8 |- ((x = B /\ (ph /\ B e. A /\ x e. A)) -> ps)
11	3, 4, 5, 8, 10	syl13anc 1102	. . . . . . 7 |- (((ph /\ B e. A) /\ x = B) -> ps)
12		sbceq1a 2456	. . . . . . . 8 |- (x = B -> (ps <-> [B / x]ps))
13	12	adantl 424	. . . . . . 7 |- (((ph /\ B e. A) /\ x = B) -> (ps <-> [B / x]ps))
14	11, 13	mpbid 212	. . . . . 6 |- (((ph /\ B e. A) /\ x = B) -> [B / x]ps)
15	14	ex 402	. . . . 5 |- ((ph /\ B e. A) -> (x = B -> [B / x]ps))
16	15	19.21aiv 1664	. . . 4 |- ((ph /\ B e. A) -> A.x(x = B -> [B / x]ps))
17		ax-17 1317	. . . . . . . 8 |- (y e. B -> A.x y e. B)
18	17	hbsbc1g 2461	. . . . . . 7 |- (B e. A -> ([B / x]ps -> A.x[B / x]ps))
19	18	19.21aiv 1664	. . . . . 6 |- (B e. A -> A.x([B / x]ps -> A.x[B / x]ps))
20	19	adantl 424	. . . . 5 |- ((ph /\ B e. A) -> A.x([B / x]ps -> A.x[B / x]ps))
21		19.23t 1474	. . . . 5 |- (A.x([B / x]ps -> A.x[B / x]ps) -> (A.x(x = B -> [B / x]ps) <-> (E.x x = B -> [B / x]ps)))
22	20, 21	syl 12	. . . 4 |- ((ph /\ B e. A) -> (A.x(x = B -> [B / x]ps) <-> (E.x x = B -> [B / x]ps)))
23	16, 22	mpbid 212	. . 3 |- ((ph /\ B e. A) -> (E.x x = B -> [B / x]ps))
24	2, 23	mpd 29	. 2 |- ((ph /\ B e. A) -> [B / x]ps)
25		simpr 350	. . 3 |- ((ph /\ B e. A) -> B e. A)
26	9	3expa 1067	. . . . . 6 |- (((ph /\ B e. A) /\ x e. A) -> (ps <-> x = B))
27	26	r19.21aiva 2176	. . . . 5 |- ((ph /\ B e. A) -> A.x e. A (ps <-> x = B))
28		eqeq2 1893	. . . . . . . 8 |- (y = B -> (x = y <-> x = B))
29	28	bibi2d 680	. . . . . . 7 |- (y = B -> ((ps <-> x = y) <-> (ps <-> x = B)))
30	29	ralbidv 2123	. . . . . 6 |- (y = B -> (A.x e. A (ps <-> x = y) <-> A.x e. A (ps <-> x = B)))
31	30	rcla4ev 2381	. . . . 5 |- ((B e. A /\ A.x e. A (ps <-> x = B)) -> E.y e. A A.x e. A (ps <-> x = y))
32	25, 27, 31	syl11anc 524	. . . 4 |- ((ph /\ B e. A) -> E.y e. A A.x e. A (ps <-> x = y))
33		reu6 2443	. . . 4 |- (E!x e. A ps <-> E.y e. A A.x e. A (ps <-> x = y))
34	32, 33	sylibr 217	. . 3 |- ((ph /\ B e. A) -> E!x e. A ps)
35	17, 18, 12	riota2f 5579	. . 3 |- ((B e. A /\ E!x e. A ps) -> ([B / x]ps <-> (iota_x e. Aps) = B))
36	25, 34, 35	syl11anc 524	. 2 |- ((ph /\ B e. A) -> ([B / x]ps <-> (iota_x e. Aps) = B))
37	24, 36	mpbid 212	1 |- ((ph /\ B e. A) -> (iota_x e. Aps) = B)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  A.wral 2105  E.wrex 2106  E!wreu 2107  iota_crio 5555

This theorem is referenced by:  lubid 16807  lubun 16899  lubunNEW 16900  glb0 16920

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-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-reu 2111  df-v 2294  df-sbc 2454  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-uni 3178  df-iota 5089  df-riota 5560

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