Theorem riotass 5438

Description: Restriction of a unique element to a smaller class. (Contributed by NM, 19-Oct-2005.) (Revised by Mario Carneiro, 24-Dec-2016.)

Assertion

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

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   ph(x)

Proof of Theorem riotass

Step	Hyp	Ref	Expression
1		reuss 3212	. . . 4 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> E!x e. A ph)
2		riotasbc 5426	. . . 4 |- (E!x e. A ph -> [.( iota_x e. A ph) / x ].ph)
3	1, 2	syl 14	. . 3 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> [.( iota_x e. A ph) / x ].ph)
4		simp1 903	. . . . 5 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> A C_ B)
5		riotacl 5425	. . . . . 6 |- (E!x e. A ph -> ( iota_x e. A ph) e. A)
6	1, 5	syl 14	. . . . 5 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> ( iota_x e. A ph) e. A)
7	4, 6	sseldd 2940	. . . 4 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> ( iota_x e. A ph) e. B)
8		simp3 905	. . . 4 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> E!x e. B ph)
9		nfriota1 5418	. . . . 5 |- F/_x( iota_x e. A ph)
10	9	nfsbc1 2775	. . . . 5 |- F/x [.( iota_x e. A ph) / x ].ph
11		sbceq1a 2767	. . . . 5 |- (x = ( iota_x e. A ph) -> (ph <-> [.( iota_x e. A ph) / x ].ph))
12	9, 10, 11	riota2f 5432	. . . 4 |- ((( iota_x e. A ph) e. B /\ E!x e. B ph) -> ( [.( iota_x e. A ph) / x ].ph <-> ( iota_x e. B ph) = ( iota_x e. A ph)))
13	7, 8, 12	syl2anc 391	. . 3 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> ( [.( iota_x e. A ph) / x ].ph <-> ( iota_x e. B ph) = ( iota_x e. A ph)))
14	3, 13	mpbid 135	. 2 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> ( iota_x e. B ph) = ( iota_x e. A ph))
15	14	eqcomd 2042	1 |- ((A C_ B /\ E.x e. A ph /\ E!x e. B ph) -> ( iota_x e. A ph) = ( iota_x e. B ph))

Syntax hints:   -> wi 4   <-> wb 98   /\ w3a 884   = wceq 1242   e. wcel 1390  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-bnd 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:  moriotass  5439