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Theorem mapsspm 8899
Description: Set exponentiation is a subset of partial maps. (Contributed by NM, 15-Nov-2007.) (Revised by Mario Carneiro, 27-Feb-2016.)
Assertion
Ref Expression
mapsspm (𝐴m 𝐵) ⊆ (𝐴pm 𝐵)

Proof of Theorem mapsspm
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 elmapex 8871 . . . 4 (𝑓 ∈ (𝐴m 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
21simprd 494 . . 3 (𝑓 ∈ (𝐴m 𝐵) → 𝐵 ∈ V)
31simpld 493 . . 3 (𝑓 ∈ (𝐴m 𝐵) → 𝐴 ∈ V)
4 elmapi 8872 . . 3 (𝑓 ∈ (𝐴m 𝐵) → 𝑓:𝐵𝐴)
5 fpmg 8891 . . 3 ((𝐵 ∈ V ∧ 𝐴 ∈ V ∧ 𝑓:𝐵𝐴) → 𝑓 ∈ (𝐴pm 𝐵))
62, 3, 4, 5syl3anc 1368 . 2 (𝑓 ∈ (𝐴m 𝐵) → 𝑓 ∈ (𝐴pm 𝐵))
76ssriv 3984 1 (𝐴m 𝐵) ⊆ (𝐴pm 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  Vcvv 3471  wss 3947  wf 6547  (class class class)co 7424  m cmap 8849  pm cpm 8850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pow 5367  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 7997  df-2nd 7998  df-map 8851  df-pm 8852
This theorem is referenced by:  mapsspw  8901  wunmap  10755  dvntaylp  26324  taylthlem1  26326  taylthlem2  26327  taylthlem2OLD  26328  mrsubrn  35128  mrsubff1  35129  msubrn  35144  msubff1  35171
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