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Theorem sbthlem2 7535
Description: Lemma for sbth 7544. (Contributed by NM, 22-Mar-1998.)
Hypotheses
Ref Expression
sbthlem.1  |-  A  e. 
_V
sbthlem.2  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
Assertion
Ref Expression
sbthlem2  |-  ( ran  g  C_  A  ->  ( A  \  ( g
" ( B  \ 
( f " U. D ) ) ) )  C_  U. D )
Distinct variable groups:    x, A    x, B    x, D    x, f    x, g
Allowed substitution hints:    A( f, g)    B( f, g)    D( f, g)

Proof of Theorem sbthlem2
StepHypRef Expression
1 sbthlem.1 . . . . . . . . 9  |-  A  e. 
_V
2 sbthlem.2 . . . . . . . . 9  |-  D  =  { x  |  ( x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) ) }
31, 2sbthlem1 7534 . . . . . . . 8  |-  U. D  C_  ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) )
4 imass2 5315 . . . . . . . 8  |-  ( U. D  C_  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) )  ->  ( f " U. D )  C_  ( f " ( A  \  ( g "
( B  \  (
f " U. D
) ) ) ) ) )
5 sscon 3601 . . . . . . . 8  |-  ( ( f " U. D
)  C_  ( f " ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) ) )  ->  ( B  \  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) )  C_  ( B  \  (
f " U. D
) ) )
63, 4, 5mp2b 10 . . . . . . 7  |-  ( B 
\  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) )  C_  ( B  \  (
f " U. D
) )
7 imass2 5315 . . . . . . 7  |-  ( ( B  \  ( f
" ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) ) ) )  C_  ( B  \  (
f " U. D
) )  ->  (
g " ( B 
\  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) 
C_  ( g "
( B  \  (
f " U. D
) ) ) )
8 sscon 3601 . . . . . . 7  |-  ( ( g " ( B 
\  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) 
C_  ( g "
( B  \  (
f " U. D
) ) )  -> 
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) )  C_  ( A  \  ( g " ( B  \  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) ) )
96, 7, 8mp2b 10 . . . . . 6  |-  ( A 
\  ( g "
( B  \  (
f " U. D
) ) ) ) 
C_  ( A  \ 
( g " ( B  \  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) )
10 imassrn 5291 . . . . . . . 8  |-  ( g
" ( B  \ 
( f " ( A  \  ( g "
( B  \  (
f " U. D
) ) ) ) ) ) )  C_  ran  g
11 sstr2 3474 . . . . . . . 8  |-  ( ( g " ( B 
\  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) 
C_  ran  g  ->  ( ran  g  C_  A  ->  ( g " ( B  \  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) 
C_  A ) )
1210, 11ax-mp 5 . . . . . . 7  |-  ( ran  g  C_  A  ->  ( g " ( B 
\  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) 
C_  A )
13 difss 3594 . . . . . . 7  |-  ( A 
\  ( g "
( B  \  (
f " U. D
) ) ) ) 
C_  A
14 ssconb 3600 . . . . . . 7  |-  ( ( ( g " ( B  \  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) 
C_  A  /\  ( A  \  ( g "
( B  \  (
f " U. D
) ) ) ) 
C_  A )  -> 
( ( g "
( B  \  (
f " ( A 
\  ( g "
( B  \  (
f " U. D
) ) ) ) ) ) )  C_  ( A  \  ( A  \  ( g "
( B  \  (
f " U. D
) ) ) ) )  <->  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) )  C_  ( A  \  ( g " ( B  \  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) ) ) )
1512, 13, 14sylancl 662 . . . . . 6  |-  ( ran  g  C_  A  ->  ( ( g " ( B  \  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) 
C_  ( A  \ 
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) )  <->  ( A  \  ( g " ( B  \  ( f " U. D ) ) ) )  C_  ( A  \  ( g " ( B  \  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) ) ) )
169, 15mpbiri 233 . . . . 5  |-  ( ran  g  C_  A  ->  ( g " ( B 
\  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) 
C_  ( A  \ 
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) )
1716, 13jctil 537 . . . 4  |-  ( ran  g  C_  A  ->  ( ( A  \  (
g " ( B 
\  ( f " U. D ) ) ) )  C_  A  /\  ( g " ( B  \  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) 
C_  ( A  \ 
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) )
181, 13ssexi 4548 . . . . 5  |-  ( A 
\  ( g "
( B  \  (
f " U. D
) ) ) )  e.  _V
19 sseq1 3488 . . . . . 6  |-  ( x  =  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) )  ->  ( x  C_  A  <->  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) )  C_  A )
)
20 imaeq2 5276 . . . . . . . . 9  |-  ( x  =  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) )  ->  ( f " x )  =  ( f " ( A  \  ( g "
( B  \  (
f " U. D
) ) ) ) ) )
2120difeq2d 3585 . . . . . . . 8  |-  ( x  =  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) )  ->  ( B  \  ( f " x
) )  =  ( B  \  ( f
" ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) ) ) ) )
2221imaeq2d 5280 . . . . . . 7  |-  ( x  =  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) )  ->  ( g " ( B  \ 
( f " x
) ) )  =  ( g " ( B  \  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) )
23 difeq2 3579 . . . . . . 7  |-  ( x  =  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) )  ->  ( A  \  x )  =  ( A  \  ( A 
\  ( g "
( B  \  (
f " U. D
) ) ) ) ) )
2422, 23sseq12d 3496 . . . . . 6  |-  ( x  =  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) )  ->  ( (
g " ( B 
\  ( f "
x ) ) ) 
C_  ( A  \  x )  <->  ( g " ( B  \ 
( f " ( A  \  ( g "
( B  \  (
f " U. D
) ) ) ) ) ) )  C_  ( A  \  ( A  \  ( g "
( B  \  (
f " U. D
) ) ) ) ) ) )
2519, 24anbi12d 710 . . . . 5  |-  ( x  =  ( A  \ 
( g " ( B  \  ( f " U. D ) ) ) )  ->  ( (
x  C_  A  /\  ( g " ( B  \  ( f "
x ) ) ) 
C_  ( A  \  x ) )  <->  ( ( A  \  ( g "
( B  \  (
f " U. D
) ) ) ) 
C_  A  /\  (
g " ( B 
\  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) 
C_  ( A  \ 
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) )
2618, 25elab 3213 . . . 4  |-  ( ( A  \  ( g
" ( B  \ 
( f " U. D ) ) ) )  e.  { x  |  ( x  C_  A  /\  ( g "
( B  \  (
f " x ) ) )  C_  ( A  \  x ) ) }  <->  ( ( A 
\  ( g "
( B  \  (
f " U. D
) ) ) ) 
C_  A  /\  (
g " ( B 
\  ( f "
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) ) 
C_  ( A  \ 
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) ) ) ) )
2717, 26sylibr 212 . . 3  |-  ( ran  g  C_  A  ->  ( A  \  ( g
" ( B  \ 
( f " U. D ) ) ) )  e.  { x  |  ( x  C_  A  /\  ( g "
( B  \  (
f " x ) ) )  C_  ( A  \  x ) ) } )
2827, 2syl6eleqr 2553 . 2  |-  ( ran  g  C_  A  ->  ( A  \  ( g
" ( B  \ 
( f " U. D ) ) ) )  e.  D )
29 elssuni 4232 . 2  |-  ( ( A  \  ( g
" ( B  \ 
( f " U. D ) ) ) )  e.  D  -> 
( A  \  (
g " ( B 
\  ( f " U. D ) ) ) )  C_  U. D )
3028, 29syl 16 1  |-  ( ran  g  C_  A  ->  ( A  \  ( g
" ( B  \ 
( f " U. D ) ) ) )  C_  U. D )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   {cab 2439   _Vcvv 3078    \ cdif 3436    C_ wss 3439   U.cuni 4202   ran crn 4952   "cima 4954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-xp 4957  df-cnv 4959  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964
This theorem is referenced by:  sbthlem3  7536
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