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Theorem tz7.48-1 6455
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
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . 5  |-  y  e. 
_V
21elrn2 4918 . . . 4  |-  ( y  e.  ran  F  <->  E. x <. x ,  y >.  e.  F )
3 vex 2791 . . . . . . . . 9  |-  x  e. 
_V
43, 1opeldm 4882 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  F  ->  x  e. 
dom  F )
5 tz7.48.1 . . . . . . . . 9  |-  F  Fn  On
6 fndm 5343 . . . . . . . . 9  |-  ( F  Fn  On  ->  dom  F  =  On )
75, 6ax-mp 8 . . . . . . . 8  |-  dom  F  =  On
84, 7syl6eleq 2373 . . . . . . 7  |-  ( <.
x ,  y >.  e.  F  ->  x  e.  On )
98ancri 535 . . . . . 6  |-  ( <.
x ,  y >.  e.  F  ->  ( x  e.  On  /\  <. x ,  y >.  e.  F
) )
10 fnopfvb 5564 . . . . . . . 8  |-  ( ( F  Fn  On  /\  x  e.  On )  ->  ( ( F `  x )  =  y  <->  <. x ,  y >.  e.  F ) )
115, 10mpan 651 . . . . . . 7  |-  ( x  e.  On  ->  (
( F `  x
)  =  y  <->  <. x ,  y >.  e.  F
) )
1211pm5.32i 618 . . . . . 6  |-  ( ( x  e.  On  /\  ( F `  x )  =  y )  <->  ( x  e.  On  /\  <. x ,  y >.  e.  F
) )
139, 12sylibr 203 . . . . 5  |-  ( <.
x ,  y >.  e.  F  ->  ( x  e.  On  /\  ( F `  x )  =  y ) )
1413eximi 1563 . . . 4  |-  ( E. x <. x ,  y
>.  e.  F  ->  E. x
( x  e.  On  /\  ( F `  x
)  =  y ) )
152, 14sylbi 187 . . 3  |-  ( y  e.  ran  F  ->  E. x ( x  e.  On  /\  ( F `
 x )  =  y ) )
16 nfra1 2593 . . . 4  |-  F/ x A. x  e.  On  ( F `  x )  e.  ( A  \ 
( F " x
) )
17 nfv 1605 . . . 4  |-  F/ x  y  e.  A
18 rsp 2603 . . . . 5  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( x  e.  On  ->  ( F `  x )  e.  ( A  \  ( F
" x ) ) ) )
19 eldifi 3298 . . . . . . . 8  |-  ( ( F `  x )  e.  ( A  \ 
( F " x
) )  ->  ( F `  x )  e.  A )
20 eleq1 2343 . . . . . . . 8  |-  ( ( F `  x )  =  y  ->  (
( F `  x
)  e.  A  <->  y  e.  A ) )
2119, 20syl5ibcom 211 . . . . . . 7  |-  ( ( F `  x )  e.  ( A  \ 
( F " x
) )  ->  (
( F `  x
)  =  y  -> 
y  e.  A ) )
2221imim2i 13 . . . . . 6  |-  ( ( x  e.  On  ->  ( F `  x )  e.  ( A  \ 
( F " x
) ) )  -> 
( x  e.  On  ->  ( ( F `  x )  =  y  ->  y  e.  A
) ) )
2322imp3a 420 . . . . 5  |-  ( ( x  e.  On  ->  ( F `  x )  e.  ( A  \ 
( F " x
) ) )  -> 
( ( x  e.  On  /\  ( F `
 x )  =  y )  ->  y  e.  A ) )
2418, 23syl 15 . . . 4  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( ( x  e.  On  /\  ( F `  x )  =  y )  -> 
y  e.  A ) )
2516, 17, 24exlimd 1803 . . 3  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( E. x
( x  e.  On  /\  ( F `  x
)  =  y )  ->  y  e.  A
) )
2615, 25syl5 28 . 2  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( y  e. 
ran  F  ->  y  e.  A ) )
2726ssrdv 3185 1  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ran  F  C_  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   A.wral 2543    \ cdif 3149    C_ wss 3152   <.cop 3643   Oncon0 4392   dom cdm 4689   ran crn 4690   "cima 4692    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  tz7.48-3  6456
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-iota 5219  df-fun 5257  df-fn 5258  df-fv 5263
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