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Theorem tz9.12lem1 7549
Description: Lemma for tz9.12 7552. (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 5107 . 2  |-  ( F
" A )  C_  ran  F
2 tz9.12lem.2 . . . 4  |-  F  =  ( z  e.  _V  |->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) } )
32rnmpt 5007 . . 3  |-  ran  F  =  { x  |  E. z  e.  _V  x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v
) } }
4 id 19 . . . . . 6  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) } )
5 ssrab2 3334 . . . . . . 7  |-  { v  e.  On  |  z  e.  ( R1 `  v ) }  C_  On
6 vex 2867 . . . . . . . . 9  |-  x  e. 
_V
74, 6syl6eqelr 2447 . . . . . . . 8  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) }  e.  _V )
8 intex 4248 . . . . . . . 8  |-  ( { v  e.  On  | 
z  e.  ( R1
`  v ) }  =/=  (/)  <->  |^| { v  e.  On  |  z  e.  ( R1 `  v
) }  e.  _V )
97, 8sylibr 203 . . . . . . 7  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  { v  e.  On  | 
z  e.  ( R1
`  v ) }  =/=  (/) )
10 oninton 4673 . . . . . . 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 )
115, 9, 10sylancr 644 . . . . . 6  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) }  e.  On )
124, 11eqeltrd 2432 . . . . 5  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  x  e.  On )
1312rexlimivw 2739 . . . 4  |-  ( E. z  e.  _V  x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v
) }  ->  x  e.  On )
1413abssi 3324 . . 3  |-  { x  |  E. z  e.  _V  x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) } }  C_  On
153, 14eqsstri 3284 . 2  |-  ran  F  C_  On
161, 15sstri 3264 1  |-  ( F
" A )  C_  On
Colors of variables: wff set class
Syntax hints:    = wceq 1642    e. wcel 1710   {cab 2344    =/= wne 2521   E.wrex 2620   {crab 2623   _Vcvv 2864    C_ wss 3228   (/)c0 3531   |^|cint 3943    e. cmpt 4158   Oncon0 4474   ran crn 4772   "cima 4774   ` cfv 5337   R1cr1 7524
This theorem is referenced by:  tz9.12lem2  7550  tz9.12lem3  7551
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-sep 4222  ax-nul 4230  ax-pr 4295  ax-un 4594
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-rab 2628  df-v 2866  df-sbc 3068  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 3909  df-int 3944  df-br 4105  df-opab 4159  df-mpt 4160  df-tr 4195  df-eprel 4387  df-po 4396  df-so 4397  df-fr 4434  df-we 4436  df-ord 4477  df-on 4478  df-xp 4777  df-cnv 4779  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784
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