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Theorem tz9.12lem1 7669
Description: Lemma for tz9.12 7672. (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 5175 . 2  |-  ( F
" A )  C_  ran  F
2 tz9.12lem.2 . . . 4  |-  F  =  ( z  e.  _V  |->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) } )
32rnmpt 5075 . . 3  |-  ran  F  =  { x  |  E. z  e.  _V  x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v
) } }
4 id 20 . . . . . 6  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) } )
5 ssrab2 3388 . . . . . . 7  |-  { v  e.  On  |  z  e.  ( R1 `  v ) }  C_  On
6 vex 2919 . . . . . . . . 9  |-  x  e. 
_V
74, 6syl6eqelr 2493 . . . . . . . 8  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) }  e.  _V )
8 intex 4316 . . . . . . . 8  |-  ( { v  e.  On  | 
z  e.  ( R1
`  v ) }  =/=  (/)  <->  |^| { v  e.  On  |  z  e.  ( R1 `  v
) }  e.  _V )
97, 8sylibr 204 . . . . . . 7  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  { v  e.  On  | 
z  e.  ( R1
`  v ) }  =/=  (/) )
10 oninton 4739 . . . . . . 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 645 . . . . . 6  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) }  e.  On )
124, 11eqeltrd 2478 . . . . 5  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  x  e.  On )
1312rexlimivw 2786 . . . 4  |-  ( E. z  e.  _V  x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v
) }  ->  x  e.  On )
1413abssi 3378 . . 3  |-  { x  |  E. z  e.  _V  x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) } }  C_  On
153, 14eqsstri 3338 . 2  |-  ran  F  C_  On
161, 15sstri 3317 1  |-  ( F
" A )  C_  On
Colors of variables: wff set class
Syntax hints:    = wceq 1649    e. wcel 1721   {cab 2390    =/= wne 2567   E.wrex 2667   {crab 2670   _Vcvv 2916    C_ wss 3280   (/)c0 3588   |^|cint 4010    e. cmpt 4226   Oncon0 4541   ran crn 4838   "cima 4840   ` cfv 5413   R1cr1 7644
This theorem is referenced by:  tz9.12lem2  7670  tz9.12lem3  7671
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-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-rab 2675  df-v 2918  df-sbc 3122  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-int 4011  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545  df-xp 4843  df-cnv 4845  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850
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