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Theorem fvimacnvALT 5644
Description: Another proof of fvimacnv 5640, based on funimass3 5641. If funimass3 5641 is ever proved directly, as opposed to using funimacnv 5324 pointwise, then the proof of funimacnv 5324 should be replaced with this one. (Contributed by Raph Levien, 20-Nov-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
fvimacnvALT  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  <->  A  e.  ( `' F " B ) ) )

Proof of Theorem fvimacnvALT
StepHypRef Expression
1 snssi 3759 . . 3  |-  ( A  e.  dom  F  ->  { A }  C_  dom  F )
2 funimass3 5641 . . 3  |-  ( ( Fun  F  /\  { A }  C_  dom  F
)  ->  ( ( F " { A }
)  C_  B  <->  { A }  C_  ( `' F " B ) ) )
31, 2sylan2 460 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F " { A } )  C_  B 
<->  { A }  C_  ( `' F " B ) ) )
4 fvex 5539 . . . 4  |-  ( F `
 A )  e. 
_V
54snss 3748 . . 3  |-  ( ( F `  A )  e.  B  <->  { ( F `  A ) }  C_  B )
6 eqid 2283 . . . . . 6  |-  dom  F  =  dom  F
7 df-fn 5258 . . . . . . 7  |-  ( F  Fn  dom  F  <->  ( Fun  F  /\  dom  F  =  dom  F ) )
87biimpri 197 . . . . . 6  |-  ( ( Fun  F  /\  dom  F  =  dom  F )  ->  F  Fn  dom  F )
96, 8mpan2 652 . . . . 5  |-  ( Fun 
F  ->  F  Fn  dom  F )
10 fnsnfv 5582 . . . . 5  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
119, 10sylan 457 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
1211sseq1d 3205 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( { ( F `
 A ) } 
C_  B  <->  ( F " { A } ) 
C_  B ) )
135, 12syl5bb 248 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  <->  ( F " { A } )  C_  B
) )
14 snssg 3754 . . 3  |-  ( A  e.  dom  F  -> 
( A  e.  ( `' F " B )  <->  { A }  C_  ( `' F " B ) ) )
1514adantl 452 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A  e.  ( `' F " B )  <->  { A }  C_  ( `' F " B ) ) )
163, 13, 153bitr4d 276 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  <->  A  e.  ( `' F " B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684    C_ wss 3152   {csn 3640   `'ccnv 4688   dom cdm 4689   "cima 4692   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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