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Theorem mapsspm 8415
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 8402 . . . 4 (𝑓 ∈ (𝐴m 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
21simprd 499 . . 3 (𝑓 ∈ (𝐴m 𝐵) → 𝐵 ∈ V)
31simpld 498 . . 3 (𝑓 ∈ (𝐴m 𝐵) → 𝐴 ∈ V)
4 elmapi 8403 . . 3 (𝑓 ∈ (𝐴m 𝐵) → 𝑓:𝐵𝐴)
5 fpmg 8407 . . 3 ((𝐵 ∈ V ∧ 𝐴 ∈ V ∧ 𝑓:𝐵𝐴) → 𝑓 ∈ (𝐴pm 𝐵))
62, 3, 4, 5syl3anc 1368 . 2 (𝑓 ∈ (𝐴m 𝐵) → 𝑓 ∈ (𝐴pm 𝐵))
76ssriv 3947 1 (𝐴m 𝐵) ⊆ (𝐴pm 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wcel 2115  Vcvv 3471  wss 3910  wf 6324  (class class class)co 7130  m cmap 8381  pm cpm 8382
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 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
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 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-map 8383  df-pm 8384
This theorem is referenced by:  mapsspw  8417  wunmap  10125  dvntaylp  24945  taylthlem1  24947  taylthlem2  24948  mrsubrn  32768  mrsubff1  32769  msubrn  32784  msubff1  32811
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