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Theorem pmapssbaN 30558
Description: A weakening of pmapssat 30557 to shorten some proofs. (Contributed by NM, 7-Mar-2012.) (New usage is discouraged.)
Hypotheses
Ref Expression
pmapssba.b  |-  B  =  ( Base `  K
)
pmapssba.m  |-  M  =  ( pmap `  K
)
Assertion
Ref Expression
pmapssbaN  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  C_  B )

Proof of Theorem pmapssbaN
StepHypRef Expression
1 pmapssba.b . . 3  |-  B  =  ( Base `  K
)
2 eqid 2437 . . 3  |-  ( Atoms `  K )  =  (
Atoms `  K )
3 pmapssba.m . . 3  |-  M  =  ( pmap `  K
)
41, 2, 3pmapssat 30557 . 2  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  C_  ( Atoms `  K ) )
51, 2atssbase 30089 . 2  |-  ( Atoms `  K )  C_  B
64, 5syl6ss 3361 1  |-  ( ( K  e.  C  /\  X  e.  B )  ->  ( M `  X
)  C_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3321   ` cfv 5455   Basecbs 13470   Atomscatm 30062   pmapcpmap 30295
This theorem is referenced by:  paddunN  30725
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-reu 2713  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ats 30066  df-pmap 30302
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