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Theorem fvimacnvALT 5960
Description: Alternate proof of fvimacnv 5956, based on funimass3 5957. If funimass3 5957 is ever proved directly, as opposed to using funimacnv 5616 pointwise, then the proof of funimacnv 5616 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 4087 . . 3  |-  ( A  e.  dom  F  ->  { A }  C_  dom  F )
2 funimass3 5957 . . 3  |-  ( ( Fun  F  /\  { A }  C_  dom  F
)  ->  ( ( F " { A }
)  C_  B  <->  { A }  C_  ( `' F " B ) ) )
31, 2sylan2 476 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F " { A } )  C_  B 
<->  { A }  C_  ( `' F " B ) ) )
4 fvex 5835 . . . 4  |-  ( F `
 A )  e. 
_V
54snss 4067 . . 3  |-  ( ( F `  A )  e.  B  <->  { ( F `  A ) }  C_  B )
6 eqid 2428 . . . . . 6  |-  dom  F  =  dom  F
7 df-fn 5547 . . . . . . 7  |-  ( F  Fn  dom  F  <->  ( Fun  F  /\  dom  F  =  dom  F ) )
87biimpri 209 . . . . . 6  |-  ( ( Fun  F  /\  dom  F  =  dom  F )  ->  F  Fn  dom  F )
96, 8mpan2 675 . . . . 5  |-  ( Fun 
F  ->  F  Fn  dom  F )
10 fnsnfv 5885 . . . . 5  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
119, 10sylan 473 . . . 4  |-  ( ( Fun  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
1211sseq1d 3434 . . 3  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( { ( F `
 A ) } 
C_  B  <->  ( F " { A } ) 
C_  B ) )
135, 12syl5bb 260 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F `  A )  e.  B  <->  ( F " { A } )  C_  B
) )
14 snssg 4076 . . 3  |-  ( A  e.  dom  F  -> 
( A  e.  ( `' F " B )  <->  { A }  C_  ( `' F " B ) ) )
1514adantl 467 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( A  e.  ( `' F " B )  <->  { A }  C_  ( `' F " B ) ) )
163, 13, 153bitr4d 288 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 187    /\ wa 370    = wceq 1437    e. wcel 1872    C_ wss 3379   {csn 3941   `'ccnv 4795   dom cdm 4796   "cima 4799   Fun wfun 5538    Fn wfn 5539   ` cfv 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-id 4711  df-xp 4802  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-rn 4807  df-res 4808  df-ima 4809  df-iota 5508  df-fun 5546  df-fn 5547  df-fv 5552
This theorem is referenced by: (None)
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