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Theorem tz7.48-3 6540
Description: Proposition 7.48(3) of [TakeutiZaring] p. 51. (Contributed by NM, 9-Feb-1997.)
Hypothesis
Ref Expression
tz7.48.1  |-  F  Fn  On
Assertion
Ref Expression
tz7.48-3  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  -.  A  e.  _V )
Distinct variable groups:    x, F    x, A

Proof of Theorem tz7.48-3
StepHypRef Expression
1 onprc 4655 . . . 4  |-  -.  On  e.  _V
2 tz7.48.1 . . . . . 6  |-  F  Fn  On
3 fndm 5422 . . . . . 6  |-  ( F  Fn  On  ->  dom  F  =  On )
42, 3ax-mp 8 . . . . 5  |-  dom  F  =  On
54eleq1i 2421 . . . 4  |-  ( dom 
F  e.  _V  <->  On  e.  _V )
61, 5mtbir 290 . . 3  |-  -.  dom  F  e.  _V
72tz7.48-2 6538 . . . 4  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  Fun  `' F
)
8 funrnex 5830 . . . . . 6  |-  ( dom  `' F  e.  _V  ->  ( Fun  `' F  ->  ran  `' F  e. 
_V ) )
98com12 27 . . . . 5  |-  ( Fun  `' F  ->  ( dom  `' F  e.  _V  ->  ran  `' F  e. 
_V ) )
10 df-rn 4779 . . . . . 6  |-  ran  F  =  dom  `' F
1110eleq1i 2421 . . . . 5  |-  ( ran 
F  e.  _V  <->  dom  `' F  e.  _V )
12 dfdm4 4951 . . . . . 6  |-  dom  F  =  ran  `' F
1312eleq1i 2421 . . . . 5  |-  ( dom 
F  e.  _V  <->  ran  `' F  e.  _V )
149, 11, 133imtr4g 261 . . . 4  |-  ( Fun  `' F  ->  ( ran 
F  e.  _V  ->  dom 
F  e.  _V )
)
157, 14syl 15 . . 3  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( ran  F  e.  _V  ->  dom  F  e. 
_V ) )
166, 15mtoi 169 . 2  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  -.  ran  F  e. 
_V )
172tz7.48-1 6539 . . 3  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ran  F  C_  A
)
18 ssexg 4239 . . . 4  |-  ( ( ran  F  C_  A  /\  A  e.  _V )  ->  ran  F  e.  _V )
1918ex 423 . . 3  |-  ( ran 
F  C_  A  ->  ( A  e.  _V  ->  ran 
F  e.  _V )
)
2017, 19syl 15 . 2  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( A  e. 
_V  ->  ran  F  e.  _V ) )
2116, 20mtod 168 1  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  -.  A  e.  _V )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1642    e. wcel 1710   A.wral 2619   _Vcvv 2864    \ cdif 3225    C_ wss 3228   Oncon0 4471   `'ccnv 4767   dom cdm 4768   ran crn 4769   "cima 4771   Fun wfun 5328    Fn wfn 5329   ` cfv 5334
This theorem is referenced by:  tz7.49  6541
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-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4210  ax-sep 4220  ax-nul 4228  ax-pr 4293  ax-un 4591
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-reu 2626  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-pss 3244  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-uni 3907  df-iun 3986  df-br 4103  df-opab 4157  df-mpt 4158  df-tr 4193  df-eprel 4384  df-id 4388  df-po 4393  df-so 4394  df-fr 4431  df-we 4433  df-ord 4474  df-on 4475  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-rn 4779  df-res 4780  df-ima 4781  df-iota 5298  df-fun 5336  df-fn 5337  df-f 5338  df-f1 5339  df-fo 5340  df-f1o 5341  df-fv 5342
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