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Theorem pmapssbaN 37028
 Description: A weakening of pmapssat 37027 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 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) ⊆ 𝐵)

Proof of Theorem pmapssbaN
StepHypRef Expression
1 pmapssba.b . . 3 𝐵 = (Base‘𝐾)
2 eqid 2824 . . 3 (Atoms‘𝐾) = (Atoms‘𝐾)
3 pmapssba.m . . 3 𝑀 = (pmap‘𝐾)
41, 2, 3pmapssat 37027 . 2 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) ⊆ (Atoms‘𝐾))
51, 2atssbase 36558 . 2 (Atoms‘𝐾) ⊆ 𝐵
64, 5sstrdi 3965 1 ((𝐾𝐶𝑋𝐵) → (𝑀𝑋) ⊆ 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2115   ⊆ wss 3919  ‘cfv 6345  Basecbs 16485  Atomscatm 36531  pmapcpmap 36765 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fn 6348  df-f 6349  df-f1 6350  df-fo 6351  df-f1o 6352  df-fv 6353  df-ats 36535  df-pmap 36772 This theorem is referenced by:  paddunN  37195
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