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Theorem fvimacnvALT 5982
Description: Alternate proof of fvimacnv 5978, based on funimass3 5979. If funimass3 5979 is ever proved directly, as opposed to using funimacnv 5642 pointwise, then the proof of funimacnv 5642 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 4160 . . 3  |-  ( A  e.  dom  F  ->  { A }  C_  dom  F )
2 funimass3 5979 . . 3  |-  ( ( Fun  F  /\  { A }  C_  dom  F
)  ->  ( ( F " { A }
)  C_  B  <->  { A }  C_  ( `' F " B ) ) )
31, 2sylan2 472 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F " { A } )  C_  B 
<->  { A }  C_  ( `' F " B ) ) )
4 fvex 5858 . . . 4  |-  ( F `
 A )  e. 
_V
54snss 4140 . . 3  |-  ( ( F `  A )  e.  B  <->  { ( F `  A ) }  C_  B )
6 eqid 2454 . . . . . 6  |-  dom  F  =  dom  F
7 df-fn 5573 . . . . . . 7  |-  ( F  Fn  dom  F  <->  ( Fun  F  /\  dom  F  =  dom  F ) )
87biimpri 206 . . . . . 6  |-  ( ( Fun  F  /\  dom  F  =  dom  F )  ->  F  Fn  dom  F )
96, 8mpan2 669 . . . . 5  |-  ( Fun 
F  ->  F  Fn  dom  F )
10 fnsnfv 5908 . . . . 5  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
119, 10sylan 469 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
1211sseq1d 3516 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( { ( F `
 A ) } 
C_  B  <->  ( F " { A } ) 
C_  B ) )
135, 12syl5bb 257 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  <->  ( F " { A } )  C_  B
) )
14 snssg 4149 . . 3  |-  ( A  e.  dom  F  -> 
( A  e.  ( `' F " B )  <->  { A }  C_  ( `' F " B ) ) )
1514adantl 464 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A  e.  ( `' F " B )  <->  { A }  C_  ( `' F " B ) ) )
163, 13, 153bitr4d 285 1  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  <->  A  e.  ( `' F " B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823    C_ wss 3461   {csn 4016   `'ccnv 4987   dom cdm 4988   "cima 4991   Fun wfun 5564    Fn wfn 5565   ` cfv 5570
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-fv 5578
This theorem is referenced by: (None)
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