Theorem riotaxfrd 5581

Description: Change the variable x in the expression for "the unique A such that ph " to another variable y contained in expression B. Use reuhypd 3848 to eliminate the last hypothesis. (Th. reuunixfr 3850 analog.)

Hypotheses

Ref	Expression
riotaxfrd.1	|- (z e. C -> A.y z e. C)
riotaxfrd.2	|- ((ph /\ y e. A) -> B e. A)
riotaxfrd.3	|- ((ph /\ (iota_y e. Ach) e. A) -> C e. A)
riotaxfrd.4	|- (x = B -> (ps <-> ch))
riotaxfrd.5	|- (y = (iota_y e. Ach) -> B = C)
riotaxfrd.6	|- ((ph /\ x e. A) -> E!y e. A x = B)

Assertion

Ref	Expression
riotaxfrd	|- ((ph /\ E!x e. A ps) -> (iota_x e. Aps) = C)

Distinct variable groups:   x,B   x,z,C   x,y,A,z   ph,x,y   ps,y,z   ch,x,z

Proof of Theorem riotaxfrd

Step	Hyp	Ref	Expression
1		riotaxfrd.2	. . . . . 6 |- ((ph /\ y e. A) -> B e. A)
2		riotaxfrd.6	. . . . . 6 |- ((ph /\ x e. A) -> E!y e. A x = B)
3		riotaxfrd.4	. . . . . 6 |- (x = B -> (ps <-> ch))
4	1, 2, 3	reuxfrd 3846	. . . . 5 |- (ph -> (E!x e. A ps <-> E!y e. A ch))
5		riotacl2 5578	. . . . . . . 8 |- (E!y e. A ch -> (iota_y e. Ach) e. {y e. A | ch})
6	5	adantl 424	. . . . . . 7 |- ((ph /\ E!y e. A ch) -> (iota_y e. Ach) e. {y e. A | ch})
7		hbriota1 5569	. . . . . . . . 9 |- (z e. (iota_y e. Ach) -> A.y z e. (iota_y e. Ach))
8		riotaxfrd.1	. . . . . . . . 9 |- (z e. C -> A.y z e. C)
9		riotaxfrd.5	. . . . . . . . 9 |- (y = (iota_y e. Ach) -> B = C)
10	7, 8, 1, 3, 9	rabxfrd 3842	. . . . . . . 8 |- ((ph /\ (iota_y e. Ach) e. A) -> (C e. {x e. A | ps} <-> (iota_y e. Ach) e. {y e. A | ch}))
11		riotacl 5571	. . . . . . . 8 |- (E!y e. A ch -> (iota_y e. Ach) e. A)
12	10, 11	sylan2 500	. . . . . . 7 |- ((ph /\ E!y e. A ch) -> (C e. {x e. A | ps} <-> (iota_y e. Ach) e. {y e. A | ch}))
13	6, 12	mpbird 213	. . . . . 6 |- ((ph /\ E!y e. A ch) -> C e. {x e. A | ps})
14	13	ex 402	. . . . 5 |- (ph -> (E!y e. A ch -> C e. {x e. A | ps}))
15	4, 14	sylbid 220	. . . 4 |- (ph -> (E!x e. A ps -> C e. {x e. A | ps}))
16	15	imp 377	. . 3 |- ((ph /\ E!x e. A ps) -> C e. {x e. A | ps})
17		riotaxfrd.3	. . . . . . . 8 |- ((ph /\ (iota_y e. Ach) e. A) -> C e. A)
18	17	ex 402	. . . . . . 7 |- (ph -> ((iota_y e. Ach) e. A -> C e. A))
19	18, 11	syl5 20	. . . . . 6 |- (ph -> (E!y e. A ch -> C e. A))
20	4, 19	sylbid 220	. . . . 5 |- (ph -> (E!x e. A ps -> C e. A))
21	20	imp 377	. . . 4 |- ((ph /\ E!x e. A ps) -> C e. A)
22		rabid 2253	. . . . . . . 8 |- (x e. {x e. A | ps} <-> (x e. A /\ ps))
23	22	baibr 750	. . . . . . 7 |- (x e. A -> (ps <-> x e. {x e. A | ps}))
24	23	reubiia 2261	. . . . . 6 |- (E!x e. A ps <-> E!x e. A x e. {x e. A | ps})
25	24	biimpi 168	. . . . 5 |- (E!x e. A ps -> E!x e. A x e. {x e. A | ps})
26	25	adantl 424	. . . 4 |- ((ph /\ E!x e. A ps) -> E!x e. A x e. {x e. A | ps})
27		ax-17 1317	. . . . 5 |- (z e. C -> A.x z e. C)
28		hbrab1 2257	. . . . . . 7 |- (z e. {x e. A | ps} -> A.x z e. {x e. A | ps})
29	27, 28	hbel 1996	. . . . . 6 |- (C e. {x e. A | ps} -> A.x C e. {x e. A | ps})
30	29	a1i 8	. . . . 5 |- (C e. A -> (C e. {x e. A | ps} -> A.x C e. {x e. A | ps}))
31		eleq1 1957	. . . . 5 |- (x = C -> (x e. {x e. A | ps} <-> C e. {x e. A | ps}))
32	27, 30, 31	riota2f 5579	. . . 4 |- ((C e. A /\ E!x e. A x e. {x e. A | ps}) -> (C e. {x e. A | ps} <-> (iota_x e. Ax e. {x e. A | ps}) = C))
33	21, 26, 32	syl11anc 524	. . 3 |- ((ph /\ E!x e. A ps) -> (C e. {x e. A | ps} <-> (iota_x e. Ax e. {x e. A | ps}) = C))
34	16, 33	mpbid 212	. 2 |- ((ph /\ E!x e. A ps) -> (iota_x e. Ax e. {x e. A | ps}) = C)
35	22	baib 749	. . 3 |- (x e. A -> (x e. {x e. A | ps} <-> ps))
36	35	riotabiia 5576	. 2 |- (iota_x e. Ax e. {x e. A | ps}) = (iota_x e. Aps)
37	34, 36	syl5eqr 1942	1 |- ((ph /\ E!x e. A ps) -> (iota_x e. Aps) = C)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E!wreu 2107  {crab 2108  iota_crio 5555

This theorem is referenced by:  riotaoc 16936

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-reu 2111  df-rab 2112  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-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-iota 5089  df-riota 5560

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