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Theorem tz9.12lem1 8222
Description: Lemma for tz9.12 8225. (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 5358 . 2  |-  ( F
" A )  C_  ran  F
2 tz9.12lem.2 . . . 4  |-  F  =  ( z  e.  _V  |->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) } )
32rnmpt 5258 . . 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 3581 . . . . . . 7  |-  { v  e.  On  |  z  e.  ( R1 `  v ) }  C_  On
6 eqvisset 3117 . . . . . . . 8  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) }  e.  _V )
7 intex 4612 . . . . . . . 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 6634 . . . . . . 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 663 . . . . . 6  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) }  e.  On )
114, 10eqeltrd 2545 . . . . 5  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  x  e.  On )
1211rexlimivw 2946 . . . 4  |-  ( E. z  e.  _V  x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v
) }  ->  x  e.  On )
1312abssi 3571 . . 3  |-  { x  |  E. z  e.  _V  x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) } }  C_  On
143, 13eqsstri 3529 . 2  |-  ran  F  C_  On
151, 14sstri 3508 1  |-  ( F
" A )  C_  On
Colors of variables: wff setvar class
Syntax hints:    = wceq 1395    e. wcel 1819   {cab 2442    =/= wne 2652   E.wrex 2808   {crab 2811   _Vcvv 3109    C_ wss 3471   (/)c0 3793   |^|cint 4288    |-> cmpt 4515   Oncon0 4887   ran crn 5009   "cima 5011   ` cfv 5594   R1cr1 8197
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-xp 5014  df-cnv 5016  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021
This theorem is referenced by:  tz9.12lem2  8223  tz9.12lem3  8224
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