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Theorem isf34lem7 8750
Description: Lemma for isfin3-4 8753. (Contributed by Stefan O'Rear, 7-Nov-2014.)
Hypothesis
Ref Expression
compss.a  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
Assertion
Ref Expression
isf34lem7  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A  /\  A. y  e.  om  ( G `  y )  C_  ( G `  suc  y ) )  ->  U. ran  G  e.  ran  G )
Distinct variable groups:    x, y, A    y, F    y, G
Allowed substitution hints:    F( x)    G( x)

Proof of Theorem isf34lem7
StepHypRef Expression
1 compss.a . . . . . . 7  |-  F  =  ( x  e.  ~P A  |->  ( A  \  x ) )
21isf34lem2 8744 . . . . . 6  |-  ( A  e. FinIII  ->  F : ~P A
--> ~P A )
32adantr 463 . . . . 5  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  ->  F : ~P A --> ~P A
)
433adant3 1014 . . . 4  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A  /\  A. y  e.  om  ( G `  y )  C_  ( G `  suc  y ) )  ->  F : ~P A --> ~P A
)
5 ffn 5713 . . . 4  |-  ( F : ~P A --> ~P A  ->  F  Fn  ~P A
)
64, 5syl 16 . . 3  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A  /\  A. y  e.  om  ( G `  y )  C_  ( G `  suc  y ) )  ->  F  Fn  ~P A
)
7 imassrn 5336 . . . 4  |-  ( F
" ran  G )  C_ 
ran  F
8 frn 5719 . . . . . 6  |-  ( F : ~P A --> ~P A  ->  ran  F  C_  ~P A )
93, 8syl 16 . . . . 5  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  ->  ran  F  C_  ~P A
)
1093adant3 1014 . . . 4  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A  /\  A. y  e.  om  ( G `  y )  C_  ( G `  suc  y ) )  ->  ran  F  C_  ~P A
)
117, 10syl5ss 3500 . . 3  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A  /\  A. y  e.  om  ( G `  y )  C_  ( G `  suc  y ) )  -> 
( F " ran  G )  C_  ~P A
)
12 simp1 994 . . . . 5  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A  /\  A. y  e.  om  ( G `  y )  C_  ( G `  suc  y ) )  ->  A  e. FinIII )
13 fco 5723 . . . . . . 7  |-  ( ( F : ~P A --> ~P A  /\  G : om
--> ~P A )  -> 
( F  o.  G
) : om --> ~P A
)
142, 13sylan 469 . . . . . 6  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  -> 
( F  o.  G
) : om --> ~P A
)
15143adant3 1014 . . . . 5  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A  /\  A. y  e.  om  ( G `  y )  C_  ( G `  suc  y ) )  -> 
( F  o.  G
) : om --> ~P A
)
16 sscon 3624 . . . . . . . 8  |-  ( ( G `  y ) 
C_  ( G `  suc  y )  ->  ( A  \  ( G `  suc  y ) )  C_  ( A  \  ( G `  y )
) )
17 simpr 459 . . . . . . . . . . 11  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  ->  G : om --> ~P A
)
18 peano2 6693 . . . . . . . . . . 11  |-  ( y  e.  om  ->  suc  y  e.  om )
19 fvco3 5925 . . . . . . . . . . 11  |-  ( ( G : om --> ~P A  /\  suc  y  e.  om )  ->  ( ( F  o.  G ) `  suc  y )  =  ( F `  ( G `
 suc  y )
) )
2017, 18, 19syl2an 475 . . . . . . . . . 10  |-  ( ( ( A  e. FinIII  /\  G : om --> ~P A )  /\  y  e.  om )  ->  ( ( F  o.  G ) `  suc  y )  =  ( F `  ( G `
 suc  y )
) )
21 simpll 751 . . . . . . . . . . 11  |-  ( ( ( A  e. FinIII  /\  G : om --> ~P A )  /\  y  e.  om )  ->  A  e. FinIII )
22 ffvelrn 6005 . . . . . . . . . . . . 13  |-  ( ( G : om --> ~P A  /\  suc  y  e.  om )  ->  ( G `  suc  y )  e.  ~P A )
2317, 18, 22syl2an 475 . . . . . . . . . . . 12  |-  ( ( ( A  e. FinIII  /\  G : om --> ~P A )  /\  y  e.  om )  ->  ( G `  suc  y )  e.  ~P A )
2423elpwid 4009 . . . . . . . . . . 11  |-  ( ( ( A  e. FinIII  /\  G : om --> ~P A )  /\  y  e.  om )  ->  ( G `  suc  y )  C_  A
)
251isf34lem1 8743 . . . . . . . . . . 11  |-  ( ( A  e. FinIII  /\  ( G `  suc  y )  C_  A )  ->  ( F `  ( G `  suc  y ) )  =  ( A  \ 
( G `  suc  y ) ) )
2621, 24, 25syl2anc 659 . . . . . . . . . 10  |-  ( ( ( A  e. FinIII  /\  G : om --> ~P A )  /\  y  e.  om )  ->  ( F `  ( G `  suc  y
) )  =  ( A  \  ( G `
 suc  y )
) )
2720, 26eqtrd 2495 . . . . . . . . 9  |-  ( ( ( A  e. FinIII  /\  G : om --> ~P A )  /\  y  e.  om )  ->  ( ( F  o.  G ) `  suc  y )  =  ( A  \  ( G `
 suc  y )
) )
28 fvco3 5925 . . . . . . . . . . 11  |-  ( ( G : om --> ~P A  /\  y  e.  om )  ->  ( ( F  o.  G ) `  y )  =  ( F `  ( G `
 y ) ) )
2928adantll 711 . . . . . . . . . 10  |-  ( ( ( A  e. FinIII  /\  G : om --> ~P A )  /\  y  e.  om )  ->  ( ( F  o.  G ) `  y )  =  ( F `  ( G `
 y ) ) )
30 ffvelrn 6005 . . . . . . . . . . . . 13  |-  ( ( G : om --> ~P A  /\  y  e.  om )  ->  ( G `  y )  e.  ~P A )
3130adantll 711 . . . . . . . . . . . 12  |-  ( ( ( A  e. FinIII  /\  G : om --> ~P A )  /\  y  e.  om )  ->  ( G `  y )  e.  ~P A )
3231elpwid 4009 . . . . . . . . . . 11  |-  ( ( ( A  e. FinIII  /\  G : om --> ~P A )  /\  y  e.  om )  ->  ( G `  y )  C_  A
)
331isf34lem1 8743 . . . . . . . . . . 11  |-  ( ( A  e. FinIII  /\  ( G `  y )  C_  A
)  ->  ( F `  ( G `  y
) )  =  ( A  \  ( G `
 y ) ) )
3421, 32, 33syl2anc 659 . . . . . . . . . 10  |-  ( ( ( A  e. FinIII  /\  G : om --> ~P A )  /\  y  e.  om )  ->  ( F `  ( G `  y ) )  =  ( A 
\  ( G `  y ) ) )
3529, 34eqtrd 2495 . . . . . . . . 9  |-  ( ( ( A  e. FinIII  /\  G : om --> ~P A )  /\  y  e.  om )  ->  ( ( F  o.  G ) `  y )  =  ( A  \  ( G `
 y ) ) )
3627, 35sseq12d 3518 . . . . . . . 8  |-  ( ( ( A  e. FinIII  /\  G : om --> ~P A )  /\  y  e.  om )  ->  ( ( ( F  o.  G ) `
 suc  y )  C_  ( ( F  o.  G ) `  y
)  <->  ( A  \ 
( G `  suc  y ) )  C_  ( A  \  ( G `  y )
) ) )
3716, 36syl5ibr 221 . . . . . . 7  |-  ( ( ( A  e. FinIII  /\  G : om --> ~P A )  /\  y  e.  om )  ->  ( ( G `
 y )  C_  ( G `  suc  y
)  ->  ( ( F  o.  G ) `  suc  y )  C_  ( ( F  o.  G ) `  y
) ) )
3837ralimdva 2862 . . . . . 6  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  -> 
( A. y  e. 
om  ( G `  y )  C_  ( G `  suc  y )  ->  A. y  e.  om  ( ( F  o.  G ) `  suc  y )  C_  (
( F  o.  G
) `  y )
) )
39383impia 1191 . . . . 5  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A  /\  A. y  e.  om  ( G `  y )  C_  ( G `  suc  y ) )  ->  A. y  e.  om  ( ( F  o.  G ) `  suc  y )  C_  (
( F  o.  G
) `  y )
)
40 fin33i 8740 . . . . 5  |-  ( ( A  e. FinIII  /\  ( F  o.  G ) : om --> ~P A  /\  A. y  e.  om  ( ( F  o.  G ) `  suc  y )  C_  (
( F  o.  G
) `  y )
)  ->  |^| ran  ( F  o.  G )  e.  ran  ( F  o.  G ) )
4112, 15, 39, 40syl3anc 1226 . . . 4  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A  /\  A. y  e.  om  ( G `  y )  C_  ( G `  suc  y ) )  ->  |^| ran  ( F  o.  G )  e.  ran  ( F  o.  G
) )
42 rnco2 5497 . . . . 5  |-  ran  ( F  o.  G )  =  ( F " ran  G )
4342inteqi 4275 . . . 4  |-  |^| ran  ( F  o.  G
)  =  |^| ( F " ran  G )
4441, 43, 423eltr3g 2558 . . 3  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A  /\  A. y  e.  om  ( G `  y )  C_  ( G `  suc  y ) )  ->  |^| ( F " ran  G )  e.  ( F
" ran  G )
)
45 fnfvima 6125 . . 3  |-  ( ( F  Fn  ~P A  /\  ( F " ran  G )  C_  ~P A  /\  |^| ( F " ran  G )  e.  ( F " ran  G
) )  ->  ( F `  |^| ( F
" ran  G )
)  e.  ( F
" ( F " ran  G ) ) )
466, 11, 44, 45syl3anc 1226 . 2  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A  /\  A. y  e.  om  ( G `  y )  C_  ( G `  suc  y ) )  -> 
( F `  |^| ( F " ran  G
) )  e.  ( F " ( F
" ran  G )
) )
47 simpl 455 . . . . . 6  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  ->  A  e. FinIII )
487, 9syl5ss 3500 . . . . . 6  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  -> 
( F " ran  G )  C_  ~P A
)
49 incom 3677 . . . . . . . . 9  |-  ( dom 
F  i^i  ran  G )  =  ( ran  G  i^i  dom  F )
50 frn 5719 . . . . . . . . . . . 12  |-  ( G : om --> ~P A  ->  ran  G  C_  ~P A )
5150adantl 464 . . . . . . . . . . 11  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  ->  ran  G  C_  ~P A
)
52 fdm 5717 . . . . . . . . . . . 12  |-  ( F : ~P A --> ~P A  ->  dom  F  =  ~P A )
533, 52syl 16 . . . . . . . . . . 11  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  ->  dom  F  =  ~P A
)
5451, 53sseqtr4d 3526 . . . . . . . . . 10  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  ->  ran  G  C_  dom  F )
55 df-ss 3475 . . . . . . . . . 10  |-  ( ran 
G  C_  dom  F  <->  ( ran  G  i^i  dom  F )  =  ran  G )
5654, 55sylib 196 . . . . . . . . 9  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  -> 
( ran  G  i^i  dom 
F )  =  ran  G )
5749, 56syl5eq 2507 . . . . . . . 8  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  -> 
( dom  F  i^i  ran 
G )  =  ran  G )
58 fdm 5717 . . . . . . . . . . 11  |-  ( G : om --> ~P A  ->  dom  G  =  om )
5958adantl 464 . . . . . . . . . 10  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  ->  dom  G  =  om )
60 peano1 6692 . . . . . . . . . . 11  |-  (/)  e.  om
61 ne0i 3789 . . . . . . . . . . 11  |-  ( (/)  e.  om  ->  om  =/=  (/) )
6260, 61mp1i 12 . . . . . . . . . 10  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  ->  om  =/=  (/) )
6359, 62eqnetrd 2747 . . . . . . . . 9  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  ->  dom  G  =/=  (/) )
64 dm0rn0 5208 . . . . . . . . . 10  |-  ( dom 
G  =  (/)  <->  ran  G  =  (/) )
6564necon3bii 2722 . . . . . . . . 9  |-  ( dom 
G  =/=  (/)  <->  ran  G  =/=  (/) )
6663, 65sylib 196 . . . . . . . 8  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  ->  ran  G  =/=  (/) )
6757, 66eqnetrd 2747 . . . . . . 7  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  -> 
( dom  F  i^i  ran 
G )  =/=  (/) )
68 imadisj 5344 . . . . . . . 8  |-  ( ( F " ran  G
)  =  (/)  <->  ( dom  F  i^i  ran  G )  =  (/) )
6968necon3bii 2722 . . . . . . 7  |-  ( ( F " ran  G
)  =/=  (/)  <->  ( dom  F  i^i  ran  G )  =/=  (/) )
7067, 69sylibr 212 . . . . . 6  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  -> 
( F " ran  G )  =/=  (/) )
711isf34lem5 8749 . . . . . 6  |-  ( ( A  e. FinIII  /\  ( ( F " ran  G ) 
C_  ~P A  /\  ( F " ran  G )  =/=  (/) ) )  -> 
( F `  |^| ( F " ran  G
) )  =  U. ( F " ( F
" ran  G )
) )
7247, 48, 70, 71syl12anc 1224 . . . . 5  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  -> 
( F `  |^| ( F " ran  G
) )  =  U. ( F " ( F
" ran  G )
) )
731isf34lem3 8746 . . . . . . 7  |-  ( ( A  e. FinIII  /\  ran  G  C_  ~P A )  ->  ( F " ( F " ran  G ) )  =  ran  G )
7447, 51, 73syl2anc 659 . . . . . 6  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  -> 
( F " ( F " ran  G ) )  =  ran  G
)
7574unieqd 4245 . . . . 5  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  ->  U. ( F " ( F " ran  G ) )  =  U. ran  G )
7672, 75eqtrd 2495 . . . 4  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  -> 
( F `  |^| ( F " ran  G
) )  =  U. ran  G )
7776, 74eleq12d 2536 . . 3  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A )  -> 
( ( F `  |^| ( F " ran  G ) )  e.  ( F " ( F
" ran  G )
)  <->  U. ran  G  e. 
ran  G ) )
78773adant3 1014 . 2  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A  /\  A. y  e.  om  ( G `  y )  C_  ( G `  suc  y ) )  -> 
( ( F `  |^| ( F " ran  G ) )  e.  ( F " ( F
" ran  G )
)  <->  U. ran  G  e. 
ran  G ) )
7946, 78mpbid 210 1  |-  ( ( A  e. FinIII  /\  G : om
--> ~P A  /\  A. y  e.  om  ( G `  y )  C_  ( G `  suc  y ) )  ->  U. ran  G  e.  ran  G )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823    =/= wne 2649   A.wral 2804    \ cdif 3458    i^i cin 3460    C_ wss 3461   (/)c0 3783   ~Pcpw 3999   U.cuni 4235   |^|cint 4271    |-> cmpt 4497   suc csuc 4869   dom cdm 4988   ran crn 4989   "cima 4991    o. ccom 4992    Fn wfn 5565   -->wf 5566   ` cfv 5570   omcom 6673  FinIIIcfin3 8652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-rpss 6553  df-om 6674  df-recs 7034  df-rdg 7068  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-wdom 7977  df-card 8311  df-fin4 8658  df-fin3 8659
This theorem is referenced by:  isf34lem6  8751  fin34i  8752
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