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Theorem fvimacnvALT 5991
Description: Alternate proof of fvimacnv 5987, based on funimass3 5988. If funimass3 5988 is ever proved directly, as opposed to using funimacnv 5650 pointwise, then the proof of funimacnv 5650 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 4159 . . 3  |-  ( A  e.  dom  F  ->  { A }  C_  dom  F )
2 funimass3 5988 . . 3  |-  ( ( Fun  F  /\  { A }  C_  dom  F
)  ->  ( ( F " { A }
)  C_  B  <->  { A }  C_  ( `' F " B ) ) )
31, 2sylan2 474 . 2  |-  ( ( Fun  F  /\  A  e.  dom  F )  -> 
( ( F " { A } )  C_  B 
<->  { A }  C_  ( `' F " B ) ) )
4 fvex 5866 . . . 4  |-  ( F `
 A )  e. 
_V
54snss 4139 . . 3  |-  ( ( F `  A )  e.  B  <->  { ( F `  A ) }  C_  B )
6 eqid 2443 . . . . . 6  |-  dom  F  =  dom  F
7 df-fn 5581 . . . . . . 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 671 . . . . 5  |-  ( Fun 
F  ->  F  Fn  dom  F )
10 fnsnfv 5918 . . . . 5  |-  ( ( F  Fn  dom  F  /\  A  e.  dom  F )  ->  { ( F `  A ) }  =  ( F " { A } ) )
119, 10sylan 471 . . . 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 4148 . . 3  |-  ( A  e.  dom  F  -> 
( A  e.  ( `' F " B )  <->  { A }  C_  ( `' F " B ) ) )
1514adantl 466 . 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 369    = wceq 1383    e. wcel 1804    C_ wss 3461   {csn 4014   `'ccnv 4988   dom cdm 4989   "cima 4992   Fun wfun 5572    Fn wfn 5573   ` cfv 5578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-fv 5586
This theorem is referenced by: (None)
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