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Theorem tz7.48-1 6341
Description: Proposition 7.48(1) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)
Hypothesis
Ref Expression
tz7.48.1  |-  F  Fn  On
Assertion
Ref Expression
tz7.48-1  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ran  F  C_  A
)
Distinct variable groups:    x, F    x, A

Proof of Theorem tz7.48-1
StepHypRef Expression
1 vex 2730 . . . . 5  |-  y  e. 
_V
21elrn2 4825 . . . 4  |-  ( y  e.  ran  F  <->  E. x <. x ,  y >.  e.  F )
3 vex 2730 . . . . . . . . 9  |-  x  e. 
_V
43, 1opeldm 4789 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  F  ->  x  e. 
dom  F )
5 tz7.48.1 . . . . . . . . 9  |-  F  Fn  On
6 fndm 5200 . . . . . . . . 9  |-  ( F  Fn  On  ->  dom  F  =  On )
75, 6ax-mp 10 . . . . . . . 8  |-  dom  F  =  On
84, 7syl6eleq 2343 . . . . . . 7  |-  ( <.
x ,  y >.  e.  F  ->  x  e.  On )
98ancri 537 . . . . . 6  |-  ( <.
x ,  y >.  e.  F  ->  ( x  e.  On  /\  <. x ,  y >.  e.  F
) )
10 fnopfvb 5416 . . . . . . . 8  |-  ( ( F  Fn  On  /\  x  e.  On )  ->  ( ( F `  x )  =  y  <->  <. x ,  y >.  e.  F ) )
115, 10mpan 654 . . . . . . 7  |-  ( x  e.  On  ->  (
( F `  x
)  =  y  <->  <. x ,  y >.  e.  F
) )
1211pm5.32i 621 . . . . . 6  |-  ( ( x  e.  On  /\  ( F `  x )  =  y )  <->  ( x  e.  On  /\  <. x ,  y >.  e.  F
) )
139, 12sylibr 205 . . . . 5  |-  ( <.
x ,  y >.  e.  F  ->  ( x  e.  On  /\  ( F `  x )  =  y ) )
1413eximi 1574 . . . 4  |-  ( E. x <. x ,  y
>.  e.  F  ->  E. x
( x  e.  On  /\  ( F `  x
)  =  y ) )
152, 14sylbi 189 . . 3  |-  ( y  e.  ran  F  ->  E. x ( x  e.  On  /\  ( F `
 x )  =  y ) )
16 nfra1 2555 . . . 4  |-  F/ x A. x  e.  On  ( F `  x )  e.  ( A  \ 
( F " x
) )
17 nfv 1629 . . . 4  |-  F/ x  y  e.  A
18 ra4 2565 . . . . 5  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( x  e.  On  ->  ( F `  x )  e.  ( A  \  ( F
" x ) ) ) )
19 eldifi 3215 . . . . . . . 8  |-  ( ( F `  x )  e.  ( A  \ 
( F " x
) )  ->  ( F `  x )  e.  A )
20 eleq1 2313 . . . . . . . 8  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  e.  A  <->  y  e.  A ) )
2119, 20syl5ibcom 213 . . . . . . 7  |-  ( ( F `  x )  e.  ( A  \ 
( F " x
) )  ->  (
( F `  x
)  =  y  -> 
y  e.  A ) )
2221imim2i 15 . . . . . 6  |-  ( ( x  e.  On  ->  ( F `  x )  e.  ( A  \ 
( F " x
) ) )  -> 
( x  e.  On  ->  ( ( F `  x )  =  y  ->  y  e.  A
) ) )
2322imp3a 422 . . . . 5  |-  ( ( x  e.  On  ->  ( F `  x )  e.  ( A  \ 
( F " x
) ) )  -> 
( ( x  e.  On  /\  ( F `
 x )  =  y )  ->  y  e.  A ) )
2418, 23syl 17 . . . 4  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( ( x  e.  On  /\  ( F `  x )  =  y )  -> 
y  e.  A ) )
2516, 17, 24exlimd 1784 . . 3  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( E. x
( x  e.  On  /\  ( F `  x
)  =  y )  ->  y  e.  A
) )
2615, 25syl5 30 . 2  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( y  e. 
ran  F  ->  y  e.  A ) )
2726ssrdv 3106 1  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ran  F  C_  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   A.wral 2509    \ cdif 3075    C_ wss 3078   <.cop 3547   Oncon0 4285   dom cdm 4580   ran crn 4581   "cima 4583    Fn wfn 4587   ` cfv 4592
This theorem is referenced by:  tz7.48-3  6342
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-fv 4608
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