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Theorem tz7.48-3 6456
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 4576 . . . 4  |-  -.  On  e.  _V
2 tz7.48.1 . . . . . 6  |-  F  Fn  On
3 fndm 5343 . . . . . 6  |-  ( F  Fn  On  ->  dom  F  =  On )
42, 3ax-mp 8 . . . . 5  |-  dom  F  =  On
54eleq1i 2346 . . . 4  |-  ( dom 
F  e.  _V  <->  On  e.  _V )
61, 5mtbir 290 . . 3  |-  -.  dom  F  e.  _V
72tz7.48-2 6454 . . . 4  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  Fun  `' F
)
8 funrnex 5747 . . . . . 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 4700 . . . . . 6  |-  ran  F  =  dom  `' F
1110eleq1i 2346 . . . . 5  |-  ( ran 
F  e.  _V  <->  dom  `' F  e.  _V )
12 dfdm4 4872 . . . . . 6  |-  dom  F  =  ran  `' F
1312eleq1i 2346 . . . . 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 6455 . . 3  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ran  F  C_  A
)
18 ssexg 4160 . . . 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 1623    e. wcel 1684   A.wral 2543   _Vcvv 2788    \ cdif 3149    C_ wss 3152   Oncon0 4392   `'ccnv 4688   dom cdm 4689   ran crn 4690   "cima 4692   Fun wfun 5249    Fn wfn 5250   ` cfv 5255
This theorem is referenced by:  tz7.49  6457
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-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263
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