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Theorem nofulllem3 29681
Description: Lemma for nofull (future) . Restriction of surreal number to a superset of its birthday does not change anything. (Contributed by Scott Fenton, 25-Apr-2017.)
Assertion
Ref Expression
nofulllem3  |-  ( ( A  C_  No  /\  X  e.  A  /\  A  C_  S )  ->  ( X  |`  U. ( bday " S ) )  =  X )

Proof of Theorem nofulllem3
StepHypRef Expression
1 ssel2 3494 . . . 4  |-  ( ( A  C_  No  /\  X  e.  A )  ->  X  e.  No )
2 nofun 29626 . . . . 5  |-  ( X  e.  No  ->  Fun  X )
3 funrel 5611 . . . . 5  |-  ( Fun 
X  ->  Rel  X )
42, 3syl 16 . . . 4  |-  ( X  e.  No  ->  Rel  X )
51, 4syl 16 . . 3  |-  ( ( A  C_  No  /\  X  e.  A )  ->  Rel  X )
653adant3 1016 . 2  |-  ( ( A  C_  No  /\  X  e.  A  /\  A  C_  S )  ->  Rel  X )
7 bdayval 29625 . . . . . 6  |-  ( X  e.  No  ->  ( bday `  X )  =  dom  X )
81, 7syl 16 . . . . 5  |-  ( ( A  C_  No  /\  X  e.  A )  ->  ( bday `  X )  =  dom  X )
9 bdaydm 29655 . . . . . . . . 9  |-  dom  bday  =  No
101, 9syl6eleqr 2556 . . . . . . . 8  |-  ( ( A  C_  No  /\  X  e.  A )  ->  X  e.  dom  bday )
11 bdayfun 29653 . . . . . . . 8  |-  Fun  bday
1210, 11jctil 537 . . . . . . 7  |-  ( ( A  C_  No  /\  X  e.  A )  ->  ( Fun  bday  /\  X  e.  dom  bday ) )
13 simpr 461 . . . . . . 7  |-  ( ( A  C_  No  /\  X  e.  A )  ->  X  e.  A )
14 funfvima 6148 . . . . . . 7  |-  ( ( Fun  bday  /\  X  e. 
dom  bday )  ->  ( X  e.  A  ->  (
bday `  X )  e.  ( bday " A
) ) )
1512, 13, 14sylc 60 . . . . . 6  |-  ( ( A  C_  No  /\  X  e.  A )  ->  ( bday `  X )  e.  ( bday " A
) )
16 elssuni 4281 . . . . . 6  |-  ( (
bday `  X )  e.  ( bday " A
)  ->  ( bday `  X )  C_  U. ( bday " A ) )
1715, 16syl 16 . . . . 5  |-  ( ( A  C_  No  /\  X  e.  A )  ->  ( bday `  X )  C_  U. ( bday " A
) )
188, 17eqsstr3d 3534 . . . 4  |-  ( ( A  C_  No  /\  X  e.  A )  ->  dom  X 
C_  U. ( bday " A
) )
19183adant3 1016 . . 3  |-  ( ( A  C_  No  /\  X  e.  A  /\  A  C_  S )  ->  dom  X 
C_  U. ( bday " A
) )
20 imass2 5382 . . . . 5  |-  ( A 
C_  S  ->  ( bday " A )  C_  ( bday " S ) )
2120unissd 4275 . . . 4  |-  ( A 
C_  S  ->  U. ( bday " A )  C_  U. ( bday " S
) )
22213ad2ant3 1019 . . 3  |-  ( ( A  C_  No  /\  X  e.  A  /\  A  C_  S )  ->  U. ( bday " A )  C_  U. ( bday " S
) )
2319, 22sstrd 3509 . 2  |-  ( ( A  C_  No  /\  X  e.  A  /\  A  C_  S )  ->  dom  X 
C_  U. ( bday " S
) )
24 relssres 5321 . 2  |-  ( ( Rel  X  /\  dom  X 
C_  U. ( bday " S
) )  ->  ( X  |`  U. ( bday " S ) )  =  X )
256, 23, 24syl2anc 661 1  |-  ( ( A  C_  No  /\  X  e.  A  /\  A  C_  S )  ->  ( X  |`  U. ( bday " S ) )  =  X )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819    C_ wss 3471   U.cuni 4251   dom cdm 5008    |` cres 5010   "cima 5011   Rel wrel 5013   Fun wfun 5588   ` cfv 5594   Nocsur 29617   bdaycbday 29619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-1o 7148  df-no 29620  df-bday 29622
This theorem is referenced by:  nofulllem4  29682
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