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Theorem fvimacnvALT 5660
Description: Another proof of fvimacnv 5656, based on funimass3 5657. If funimass3 5657 is ever proved directly, as opposed to using funimacnv 5340 pointwise, then the proof of funimacnv 5340 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 3775 . . 3  |-  ( A  e.  dom  F  ->  { A }  C_  dom  F )
2 funimass3 5657 . . 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 5555 . . . 4  |-  ( F `
 A )  e. 
_V
54snss 3761 . . 3  |-  ( ( F `  A )  e.  B  <->  { ( F `  A ) }  C_  B )
6 eqid 2296 . . . . . 6  |-  dom  F  =  dom  F
7 df-fn 5274 . . . . . . 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 5598 . . . . 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 3218 . . 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 3767 . . 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 1632    e. wcel 1696    C_ wss 3165   {csn 3653   `'ccnv 4704   dom cdm 4705   "cima 4708   Fun wfun 5265    Fn wfn 5266   ` cfv 5271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-fv 5279
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