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Theorem tz7.48-1 6539
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 2867 . . . . 5  |-  y  e. 
_V
21elrn2 4997 . . . 4  |-  ( y  e.  ran  F  <->  E. x <. x ,  y >.  e.  F )
3 vex 2867 . . . . . . . . 9  |-  x  e. 
_V
43, 1opeldm 4961 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  F  ->  x  e. 
dom  F )
5 tz7.48.1 . . . . . . . . 9  |-  F  Fn  On
6 fndm 5422 . . . . . . . . 9  |-  ( F  Fn  On  ->  dom  F  =  On )
75, 6ax-mp 8 . . . . . . . 8  |-  dom  F  =  On
84, 7syl6eleq 2448 . . . . . . 7  |-  ( <.
x ,  y >.  e.  F  ->  x  e.  On )
98ancri 535 . . . . . 6  |-  ( <.
x ,  y >.  e.  F  ->  ( x  e.  On  /\  <. x ,  y >.  e.  F
) )
10 fnopfvb 5644 . . . . . . . 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 1576 . . . 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 2669 . . . 4  |-  F/ x A. x  e.  On  ( F `  x )  e.  ( A  \ 
( F " x
) )
17 nfv 1619 . . . 4  |-  F/ x  y  e.  A
18 rsp 2679 . . . . 5  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( x  e.  On  ->  ( F `  x )  e.  ( A  \  ( F
" x ) ) ) )
19 eldifi 3374 . . . . . . . 8  |-  ( ( F `  x )  e.  ( A  \ 
( F " x
) )  ->  ( F `  x )  e.  A )
20 eleq1 2418 . . . . . . . 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 1807 . . 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 3261 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 1541    = wceq 1642    e. wcel 1710   A.wral 2619    \ cdif 3225    C_ wss 3228   <.cop 3719   Oncon0 4471   dom cdm 4768   ran crn 4769   "cima 4771    Fn wfn 5329   ` cfv 5334
This theorem is referenced by:  tz7.48-3  6540
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-iota 5298  df-fun 5336  df-fn 5337  df-fv 5342
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