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Theorem tz9.12lem1 7990
Description: Lemma for tz9.12 7993. (Contributed by NM, 22-Sep-2003.) (Revised by Mario Carneiro, 11-Sep-2015.)
Hypotheses
Ref Expression
tz9.12lem.1  |-  A  e. 
_V
tz9.12lem.2  |-  F  =  ( z  e.  _V  |->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) } )
Assertion
Ref Expression
tz9.12lem1  |-  ( F
" A )  C_  On
Distinct variable group:    z, v, A
Allowed substitution hints:    F( z, v)

Proof of Theorem tz9.12lem1
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 imassrn 5177 . 2  |-  ( F
" A )  C_  ran  F
2 tz9.12lem.2 . . . 4  |-  F  =  ( z  e.  _V  |->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) } )
32rnmpt 5081 . . 3  |-  ran  F  =  { x  |  E. z  e.  _V  x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v
) } }
4 id 22 . . . . . 6  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) } )
5 ssrab2 3434 . . . . . . 7  |-  { v  e.  On  |  z  e.  ( R1 `  v ) }  C_  On
6 eqvisset 2978 . . . . . . . 8  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) }  e.  _V )
7 intex 4445 . . . . . . . 8  |-  ( { v  e.  On  | 
z  e.  ( R1
`  v ) }  =/=  (/)  <->  |^| { v  e.  On  |  z  e.  ( R1 `  v
) }  e.  _V )
86, 7sylibr 212 . . . . . . 7  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  { v  e.  On  | 
z  e.  ( R1
`  v ) }  =/=  (/) )
9 oninton 6410 . . . . . . 7  |-  ( ( { v  e.  On  |  z  e.  ( R1 `  v ) } 
C_  On  /\  { v  e.  On  |  z  e.  ( R1 `  v ) }  =/=  (/) )  ->  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  e.  On )
105, 8, 9sylancr 658 . . . . . 6  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) }  e.  On )
114, 10eqeltrd 2515 . . . . 5  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  x  e.  On )
1211rexlimivw 2835 . . . 4  |-  ( E. z  e.  _V  x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v
) }  ->  x  e.  On )
1312abssi 3424 . . 3  |-  { x  |  E. z  e.  _V  x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) } }  C_  On
143, 13eqsstri 3383 . 2  |-  ran  F  C_  On
151, 14sstri 3362 1  |-  ( F
" A )  C_  On
Colors of variables: wff setvar class
Syntax hints:    = wceq 1364    e. wcel 1761   {cab 2427    =/= wne 2604   E.wrex 2714   {crab 2717   _Vcvv 2970    C_ wss 3325   (/)c0 3634   |^|cint 4125    e. cmpt 4347   Oncon0 4715   ran crn 4837   "cima 4839   ` cfv 5415   R1cr1 7965
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-sbc 3184  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-xp 4842  df-cnv 4844  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849
This theorem is referenced by:  tz9.12lem2  7991  tz9.12lem3  7992
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