MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz7.48-3 Unicode version

Theorem tz7.48-3 6660
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 4724 . . . 4  |-  -.  On  e.  _V
2 tz7.48.1 . . . . . 6  |-  F  Fn  On
3 fndm 5503 . . . . . 6  |-  ( F  Fn  On  ->  dom  F  =  On )
42, 3ax-mp 8 . . . . 5  |-  dom  F  =  On
54eleq1i 2467 . . . 4  |-  ( dom 
F  e.  _V  <->  On  e.  _V )
61, 5mtbir 291 . . 3  |-  -.  dom  F  e.  _V
72tz7.48-2 6658 . . . 4  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  Fun  `' F
)
8 funrnex 5926 . . . . . 6  |-  ( dom  `' F  e.  _V  ->  ( Fun  `' F  ->  ran  `' F  e. 
_V ) )
98com12 29 . . . . 5  |-  ( Fun  `' F  ->  ( dom  `' F  e.  _V  ->  ran  `' F  e. 
_V ) )
10 df-rn 4848 . . . . . 6  |-  ran  F  =  dom  `' F
1110eleq1i 2467 . . . . 5  |-  ( ran 
F  e.  _V  <->  dom  `' F  e.  _V )
12 dfdm4 5022 . . . . . 6  |-  dom  F  =  ran  `' F
1312eleq1i 2467 . . . . 5  |-  ( dom 
F  e.  _V  <->  ran  `' F  e.  _V )
149, 11, 133imtr4g 262 . . . 4  |-  ( Fun  `' F  ->  ( ran 
F  e.  _V  ->  dom 
F  e.  _V )
)
157, 14syl 16 . . 3  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( ran  F  e.  _V  ->  dom  F  e. 
_V ) )
166, 15mtoi 171 . 2  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  -.  ran  F  e. 
_V )
172tz7.48-1 6659 . . 3  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ran  F  C_  A
)
18 ssexg 4309 . . . 4  |-  ( ( ran  F  C_  A  /\  A  e.  _V )  ->  ran  F  e.  _V )
1918ex 424 . . 3  |-  ( ran 
F  C_  A  ->  ( A  e.  _V  ->  ran 
F  e.  _V )
)
2017, 19syl 16 . 2  |-  ( A. x  e.  On  ( F `  x )  e.  ( A  \  ( F " x ) )  ->  ( A  e. 
_V  ->  ran  F  e.  _V ) )
2116, 20mtod 170 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 1649    e. wcel 1721   A.wral 2666   _Vcvv 2916    \ cdif 3277    C_ wss 3280   Oncon0 4541   `'ccnv 4836   dom cdm 4837   ran crn 4838   "cima 4840   Fun wfun 5407    Fn wfn 5408   ` cfv 5413
This theorem is referenced by:  tz7.49  6661
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421
  Copyright terms: Public domain W3C validator