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Theorem cantnfp1lem1 7396
Description: Lemma for cantnfp1 7399. (Contributed by Mario Carneiro, 20-Jun-2015.)
Hypotheses
Ref Expression
cantnfs.1  |-  S  =  dom  ( A CNF  B
)
cantnfs.2  |-  ( ph  ->  A  e.  On )
cantnfs.3  |-  ( ph  ->  B  e.  On )
cantnfp1.4  |-  ( ph  ->  G  e.  S )
cantnfp1.5  |-  ( ph  ->  X  e.  B )
cantnfp1.6  |-  ( ph  ->  Y  e.  A )
cantnfp1.7  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  C_  X
)
cantnfp1.f  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
Assertion
Ref Expression
cantnfp1lem1  |-  ( ph  ->  F  e.  S )
Distinct variable groups:    t, B    t, A    t, S    t, G    ph, t    t, Y   
t, X
Allowed substitution hint:    F( t)

Proof of Theorem cantnfp1lem1
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 cantnfp1.6 . . . . 5  |-  ( ph  ->  Y  e.  A )
21adantr 451 . . . 4  |-  ( (
ph  /\  t  e.  B )  ->  Y  e.  A )
3 cantnfp1.4 . . . . . . 7  |-  ( ph  ->  G  e.  S )
4 cantnfs.1 . . . . . . . 8  |-  S  =  dom  ( A CNF  B
)
5 cantnfs.2 . . . . . . . 8  |-  ( ph  ->  A  e.  On )
6 cantnfs.3 . . . . . . . 8  |-  ( ph  ->  B  e.  On )
74, 5, 6cantnfs 7383 . . . . . . 7  |-  ( ph  ->  ( G  e.  S  <->  ( G : B --> A  /\  ( `' G " ( _V 
\  1o ) )  e.  Fin ) ) )
83, 7mpbid 201 . . . . . 6  |-  ( ph  ->  ( G : B --> A  /\  ( `' G " ( _V  \  1o ) )  e.  Fin ) )
98simpld 445 . . . . 5  |-  ( ph  ->  G : B --> A )
10 ffvelrn 5679 . . . . 5  |-  ( ( G : B --> A  /\  t  e.  B )  ->  ( G `  t
)  e.  A )
119, 10sylan 457 . . . 4  |-  ( (
ph  /\  t  e.  B )  ->  ( G `  t )  e.  A )
12 ifcl 3614 . . . 4  |-  ( ( Y  e.  A  /\  ( G `  t )  e.  A )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
132, 11, 12syl2anc 642 . . 3  |-  ( (
ph  /\  t  e.  B )  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  e.  A )
14 cantnfp1.f . . 3  |-  F  =  ( t  e.  B  |->  if ( t  =  X ,  Y , 
( G `  t
) ) )
1513, 14fmptd 5700 . 2  |-  ( ph  ->  F : B --> A )
168simprd 449 . . . 4  |-  ( ph  ->  ( `' G "
( _V  \  1o ) )  e.  Fin )
17 snfi 6957 . . . 4  |-  { X }  e.  Fin
18 unfi 7140 . . . 4  |-  ( ( ( `' G "
( _V  \  1o ) )  e.  Fin  /\ 
{ X }  e.  Fin )  ->  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
)  e.  Fin )
1916, 17, 18sylancl 643 . . 3  |-  ( ph  ->  ( ( `' G " ( _V  \  1o ) )  u.  { X } )  e.  Fin )
20 df1o2 6507 . . . . . 6  |-  1o  =  { (/) }
2120difeq2i 3304 . . . . 5  |-  ( _V 
\  1o )  =  ( _V  \  { (/)
} )
2221imaeq2i 5026 . . . 4  |-  ( `' F " ( _V 
\  1o ) )  =  ( `' F " ( _V  \  { (/)
} ) )
23 eldifi 3311 . . . . . . . 8  |-  ( k  e.  ( B  \ 
( ( `' G " ( _V  \  1o ) )  u.  { X } ) )  -> 
k  e.  B )
2423adantl 452 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  k  e.  B
)
251adantr 451 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  Y  e.  A
)
26 fvex 5555 . . . . . . . 8  |-  ( G `
 k )  e. 
_V
27 ifexg 3637 . . . . . . . 8  |-  ( ( Y  e.  A  /\  ( G `  k )  e.  _V )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  e.  _V )
2825, 26, 27sylancl 643 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  e. 
_V )
29 eqeq1 2302 . . . . . . . . 9  |-  ( t  =  k  ->  (
t  =  X  <->  k  =  X ) )
30 fveq2 5541 . . . . . . . . 9  |-  ( t  =  k  ->  ( G `  t )  =  ( G `  k ) )
3129, 30ifbieq2d 3598 . . . . . . . 8  |-  ( t  =  k  ->  if ( t  =  X ,  Y ,  ( G `  t ) )  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
3231, 14fvmptg 5616 . . . . . . 7  |-  ( ( k  e.  B  /\  if ( k  =  X ,  Y ,  ( G `  k ) )  e.  _V )  ->  ( F `  k
)  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
3324, 28, 32syl2anc 642 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  ( F `  k )  =  if ( k  =  X ,  Y ,  ( G `  k ) ) )
34 eldifn 3312 . . . . . . . . 9  |-  ( k  e.  ( B  \ 
( ( `' G " ( _V  \  1o ) )  u.  { X } ) )  ->  -.  k  e.  (
( `' G "
( _V  \  1o ) )  u.  { X } ) )
3534adantl 452 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  -.  k  e.  ( ( `' G " ( _V  \  1o ) )  u.  { X } ) )
36 elsn 3668 . . . . . . . . 9  |-  ( k  e.  { X }  <->  k  =  X )
37 elun2 3356 . . . . . . . . 9  |-  ( k  e.  { X }  ->  k  e.  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
) )
3836, 37sylbir 204 . . . . . . . 8  |-  ( k  =  X  ->  k  e.  ( ( `' G " ( _V  \  1o ) )  u.  { X } ) )
3935, 38nsyl 113 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  -.  k  =  X )
40 iffalse 3585 . . . . . . 7  |-  ( -.  k  =  X  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  =  ( G `
 k ) )
4139, 40syl 15 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  if ( k  =  X ,  Y ,  ( G `  k ) )  =  ( G `  k
) )
42 ssun1 3351 . . . . . . . . 9  |-  ( `' G " ( _V 
\  1o ) ) 
C_  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
)
43 sscon 3323 . . . . . . . . 9  |-  ( ( `' G " ( _V 
\  1o ) ) 
C_  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
)  ->  ( B  \  ( ( `' G " ( _V  \  1o ) )  u.  { X } ) )  C_  ( B  \  ( `' G " ( _V 
\  1o ) ) ) )
4442, 43ax-mp 8 . . . . . . . 8  |-  ( B 
\  ( ( `' G " ( _V 
\  1o ) )  u.  { X }
) )  C_  ( B  \  ( `' G " ( _V  \  1o ) ) )
4544sseli 3189 . . . . . . 7  |-  ( k  e.  ( B  \ 
( ( `' G " ( _V  \  1o ) )  u.  { X } ) )  -> 
k  e.  ( B 
\  ( `' G " ( _V  \  1o ) ) ) )
4621imaeq2i 5026 . . . . . . . . 9  |-  ( `' G " ( _V 
\  1o ) )  =  ( `' G " ( _V  \  { (/)
} ) )
47 eqimss2 3244 . . . . . . . . 9  |-  ( ( `' G " ( _V 
\  1o ) )  =  ( `' G " ( _V  \  { (/)
} ) )  -> 
( `' G "
( _V  \  { (/)
} ) )  C_  ( `' G " ( _V 
\  1o ) ) )
4846, 47mp1i 11 . . . . . . . 8  |-  ( ph  ->  ( `' G "
( _V  \  { (/)
} ) )  C_  ( `' G " ( _V 
\  1o ) ) )
499, 48suppssr 5675 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( B  \  ( `' G " ( _V 
\  1o ) ) ) )  ->  ( G `  k )  =  (/) )
5045, 49sylan2 460 . . . . . 6  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  ( G `  k )  =  (/) )
5133, 41, 503eqtrd 2332 . . . . 5  |-  ( (
ph  /\  k  e.  ( B  \  (
( `' G "
( _V  \  1o ) )  u.  { X } ) ) )  ->  ( F `  k )  =  (/) )
5215, 51suppss 5674 . . . 4  |-  ( ph  ->  ( `' F "
( _V  \  { (/)
} ) )  C_  ( ( `' G " ( _V  \  1o ) )  u.  { X } ) )
5322, 52syl5eqss 3235 . . 3  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  C_  (
( `' G "
( _V  \  1o ) )  u.  { X } ) )
54 ssfi 7099 . . 3  |-  ( ( ( ( `' G " ( _V  \  1o ) )  u.  { X } )  e.  Fin  /\  ( `' F "
( _V  \  1o ) )  C_  (
( `' G "
( _V  \  1o ) )  u.  { X } ) )  -> 
( `' F "
( _V  \  1o ) )  e.  Fin )
5519, 53, 54syl2anc 642 . 2  |-  ( ph  ->  ( `' F "
( _V  \  1o ) )  e.  Fin )
564, 5, 6cantnfs 7383 . 2  |-  ( ph  ->  ( F  e.  S  <->  ( F : B --> A  /\  ( `' F " ( _V 
\  1o ) )  e.  Fin ) ) )
5715, 55, 56mpbir2and 888 1  |-  ( ph  ->  F  e.  S )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696   _Vcvv 2801    \ cdif 3162    u. cun 3163    C_ wss 3165   (/)c0 3468   ifcif 3578   {csn 3653    e. cmpt 4093   Oncon0 4408   `'ccnv 4704   dom cdm 4705   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874   1oc1o 6488   Fincfn 6879   CNF ccnf 7378
This theorem is referenced by:  cantnfp1lem2  7397  cantnfp1lem3  7398  cantnfp1  7399
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-riota 6320  df-recs 6404  df-rdg 6439  df-seqom 6476  df-1o 6495  df-oadd 6499  df-er 6676  df-map 6790  df-en 6880  df-fin 6883  df-oi 7241  df-cnf 7379
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