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Theorem mapsspm 8817
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 8789 . . . 4 (𝑓 ∈ (𝐴m 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
21simprd 497 . . 3 (𝑓 ∈ (𝐴m 𝐵) → 𝐵 ∈ V)
31simpld 496 . . 3 (𝑓 ∈ (𝐴m 𝐵) → 𝐴 ∈ V)
4 elmapi 8790 . . 3 (𝑓 ∈ (𝐴m 𝐵) → 𝑓:𝐵𝐴)
5 fpmg 8809 . . 3 ((𝐵 ∈ V ∧ 𝐴 ∈ V ∧ 𝑓:𝐵𝐴) → 𝑓 ∈ (𝐴pm 𝐵))
62, 3, 4, 5syl3anc 1372 . 2 (𝑓 ∈ (𝐴m 𝐵) → 𝑓 ∈ (𝐴pm 𝐵))
76ssriv 3949 1 (𝐴m 𝐵) ⊆ (𝐴pm 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wcel 2107  Vcvv 3444  wss 3911  wf 6493  (class class class)co 7358  m cmap 8768  pm cpm 8769
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-map 8770  df-pm 8771
This theorem is referenced by:  mapsspw  8819  wunmap  10667  dvntaylp  25746  taylthlem1  25748  taylthlem2  25749  mrsubrn  34164  mrsubff1  34165  msubrn  34180  msubff1  34207
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