Theorem riotass2 5437

Description: Restriction of a unique element to a smaller class. (Contributed by NM, 21-Aug-2011.) (Revised by NM, 22-Mar-2013.)

Assertion

Ref	Expression
riotass2	|- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> ( iota_x e. A ph) = ( iota_x e. B ps))

Distinct variable groups:   x,A   x,B

Allowed substitution hints:   ph(x)   ps(x)

Proof of Theorem riotass2

Step	Hyp	Ref	Expression
1		reuss2 3211	. . . 4 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> E!x e. A ph)
2		simplr 482	. . . 4 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> A.x e. A (ph -> ps))
3		riotasbc 5426	. . . . 5 |- (E!x e. A ph -> [.( iota_x e. A ph) / x ].ph)
4		riotacl 5425	. . . . . 6 |- (E!x e. A ph -> ( iota_x e. A ph) e. A)
5		rspsbc 2834	. . . . . . 7 |- (( iota_x e. A ph) e. A -> (A.x e. A (ph -> ps) -> [.( iota_x e. A ph) / x ].(ph -> ps)))
6		sbcimg 2798	. . . . . . 7 |- (( iota_x e. A ph) e. A -> ( [.( iota_x e. A ph) / x ].(ph -> ps) <-> ( [.( iota_x e. A ph) / x ].ph -> [.( iota_x e. A ph) / x ].ps)))
7	5, 6	sylibd 138	. . . . . 6 |- (( iota_x e. A ph) e. A -> (A.x e. A (ph -> ps) -> ( [.( iota_x e. A ph) / x ].ph -> [.( iota_x e. A ph) / x ].ps)))
8	4, 7	syl 14	. . . . 5 |- (E!x e. A ph -> (A.x e. A (ph -> ps) -> ( [.( iota_x e. A ph) / x ].ph -> [.( iota_x e. A ph) / x ].ps)))
9	3, 8	mpid 37	. . . 4 |- (E!x e. A ph -> (A.x e. A (ph -> ps) -> [.( iota_x e. A ph) / x ].ps))
10	1, 2, 9	sylc 56	. . 3 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> [.( iota_x e. A ph) / x ].ps)
11	1, 4	syl 14	. . . . 5 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> ( iota_x e. A ph) e. A)
12		ssel 2933	. . . . . 6 |- (A C_ B -> (( iota_x e. A ph) e. A -> ( iota_x e. A ph) e. B))
13	12	ad2antrr 457	. . . . 5 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> (( iota_x e. A ph) e. A -> ( iota_x e. A ph) e. B))
14	11, 13	mpd 13	. . . 4 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> ( iota_x e. A ph) e. B)
15		simprr 484	. . . 4 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> E!x e. B ps)
16		nfriota1 5418	. . . . 5 |- F/_x( iota_x e. A ph)
17	16	nfsbc1 2775	. . . . 5 |- F/x [.( iota_x e. A ph) / x ].ps
18		sbceq1a 2767	. . . . 5 |- (x = ( iota_x e. A ph) -> (ps <-> [.( iota_x e. A ph) / x ].ps))
19	16, 17, 18	riota2f 5432	. . . 4 |- ((( iota_x e. A ph) e. B /\ E!x e. B ps) -> ( [.( iota_x e. A ph) / x ].ps <-> ( iota_x e. B ps) = ( iota_x e. A ph)))
20	14, 15, 19	syl2anc 391	. . 3 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> ( [.( iota_x e. A ph) / x ].ps <-> ( iota_x e. B ps) = ( iota_x e. A ph)))
21	10, 20	mpbid 135	. 2 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> ( iota_x e. B ps) = ( iota_x e. A ph))
22	21	eqcomd 2042	1 |- (((A C_ B /\ A.x e. A (ph -> ps)) /\ (E.x e. A ph /\ E!x e. B ps)) -> ( iota_x e. A ph) = ( iota_x e. B ps))

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   = wceq 1242   e. wcel 1390  A.wral 2300  E.wrex 2301  E!wreu 2302   [.wsbc 2758   C_ wss 2911   iota_crio 5410

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-uni 3572  df-iota 4810  df-riota 5411

This theorem is referenced by: (None)