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Theorem tz9.12lem1 8196
Description: Lemma for tz9.12 8199. (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 5341 . 2  |-  ( F
" A )  C_  ran  F
2 tz9.12lem.2 . . . 4  |-  F  =  ( z  e.  _V  |->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) } )
32rnmpt 5241 . . 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 3580 . . . . . . 7  |-  { v  e.  On  |  z  e.  ( R1 `  v ) }  C_  On
6 eqvisset 3116 . . . . . . . 8  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  |^|
{ v  e.  On  |  z  e.  ( R1 `  v ) }  e.  _V )
7 intex 4598 . . . . . . . 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 6608 . . . . . . 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 2550 . . . . 5  |-  ( x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v ) }  ->  x  e.  On )
1211rexlimivw 2947 . . . 4  |-  ( E. z  e.  _V  x  =  |^| { v  e.  On  |  z  e.  ( R1 `  v
) }  ->  x  e.  On )
1312abssi 3570 . . 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 1374    e. wcel 1762   {cab 2447    =/= wne 2657   E.wrex 2810   {crab 2813   _Vcvv 3108    C_ wss 3471   (/)c0 3780   |^|cint 4277    |-> cmpt 4500   Oncon0 4873   ran crn 4995   "cima 4997   ` cfv 5581   R1cr1 8171
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 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681  ax-un 6569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-int 4278  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-xp 5000  df-cnv 5002  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007
This theorem is referenced by:  tz9.12lem2  8197  tz9.12lem3  8198
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