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Theorem tz7.48-1 6450
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
Dummy variable  y is distinct from all other variables.

Proof of Theorem tz7.48-1
StepHypRef Expression
1 vex 2792 . . . . 5  |-  y  e. 
_V
21elrn2 4917 . . . 4  |-  ( y  e.  ran  F  <->  E. x <. x ,  y >.  e.  F )
3 vex 2792 . . . . . . . . 9  |-  x  e. 
_V
43, 1opeldm 4881 . . . . . . . 8  |-  ( <.
x ,  y >.  e.  F  ->  x  e. 
dom  F )
5 tz7.48.1 . . . . . . . . 9  |-  F  Fn  On
6 fndm 5308 . . . . . . . . 9  |-  ( F  Fn  On  ->  dom  F  =  On )
75, 6ax-mp 10 . . . . . . . 8  |-  dom  F  =  On
84, 7syl6eleq 2374 . . . . . . 7  |-  ( <.
x ,  y >.  e.  F  ->  x  e.  On )
98ancri 537 . . . . . 6  |-  ( <.
x ,  y >.  e.  F  ->  ( x  e.  On  /\  <. x ,  y >.  e.  F
) )
10 fnopfvb 5525 . . . . . . . 8  |-  ( ( F  Fn  On  /\  x  e.  On )  ->  ( ( F `  x )  =  y  <->  <. x ,  y >.  e.  F ) )
115, 10mpan 653 . . . . . . 7  |-  ( x  e.  On  ->  (
( F `  x
)  =  y  <->  <. x ,  y >.  e.  F
) )
1211pm5.32i 620 . . . . . 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 1564 . . . 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 2594 . . . 4  |-  F/ x A. x  e.  On  ( F `  x )  e.  ( A  \ 
( F " x
) )
17 nfv 1606 . . . 4  |-  F/ x  y  e.  A
18 rsp 2604 . . . . 5  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( x  e.  On  ->  ( F `  x )  e.  ( A  \  ( F
" x ) ) ) )
19 eldifi 3299 . . . . . . . 8  |-  ( ( F `  x )  e.  ( A  \ 
( F " x
) )  ->  ( F `  x )  e.  A )
20 eleq1 2344 . . . . . . . 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 1804 . . 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 3186 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 1529    = wceq 1624    e. wcel 1685   A.wral 2544    \ cdif 3150    C_ wss 3153   <.cop 3644   Oncon0 4391   dom cdm 4688   ran crn 4689   "cima 4691    Fn wfn 5216   ` cfv 5221
This theorem is referenced by:  tz7.48-3  6451
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-fv 5229
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