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Theorem riotass 6207
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
StepHypRef Expression
1 reuss 3721 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  A  ph )
2 riotasbc 6195 . . . 4  |-  ( E! x  e.  A  ph  ->  [. ( iota_ x  e.  A  ph )  /  x ]. ph )
31, 2syl 16 . . 3  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  [. ( iota_ x  e.  A  ph )  /  x ]. ph )
4 simp1 994 . . . . 5  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  A  C_  B
)
5 riotacl 6194 . . . . . 6  |-  ( E! x  e.  A  ph  ->  ( iota_ x  e.  A  ph )  e.  A )
61, 5syl 16 . . . . 5  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  A  ph )  e.  A )
74, 6sseldd 3435 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  A  ph )  e.  B )
8 simp3 996 . . . 4  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  E! x  e.  B  ph )
9 nfriota1 6187 . . . . 5  |-  F/_ x
( iota_ x  e.  A  ph )
109nfsbc1 3288 . . . . 5  |-  F/ x [. ( iota_ x  e.  A  ph )  /  x ]. ph
11 sbceq1a 3280 . . . . 5  |-  ( x  =  ( iota_ x  e.  A  ph )  -> 
( ph  <->  [. ( iota_ x  e.  A  ph )  /  x ]. ph ) )
129, 10, 11riota2f 6201 . . . 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 )
) )
137, 8, 12syl2anc 659 . . 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 )
) )
143, 13mpbid 210 . 2  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  B  ph )  =  ( iota_ x  e.  A  ph ) )
1514eqcomd 2404 1  |-  ( ( A  C_  B  /\  E. x  e.  A  ph  /\  E! x  e.  B  ph )  ->  ( iota_ x  e.  A  ph )  =  ( iota_ x  e.  B  ph ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 971    = wceq 1399    e. wcel 1836   E.wrex 2747   E!wreu 2748   [.wsbc 3269    C_ wss 3406   iota_crio 6179
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ral 2751  df-rex 2752  df-reu 2753  df-rab 2755  df-v 3053  df-sbc 3270  df-un 3411  df-in 3413  df-ss 3420  df-sn 3962  df-pr 3964  df-uni 4181  df-iota 5477  df-riota 6180
This theorem is referenced by:  moriotass  6208
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