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Theorem mapsspm 8869
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 8841 . . . 4 (𝑓 ∈ (𝐴m 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
21simprd 495 . . 3 (𝑓 ∈ (𝐴m 𝐵) → 𝐵 ∈ V)
31simpld 494 . . 3 (𝑓 ∈ (𝐴m 𝐵) → 𝐴 ∈ V)
4 elmapi 8842 . . 3 (𝑓 ∈ (𝐴m 𝐵) → 𝑓:𝐵𝐴)
5 fpmg 8861 . . 3 ((𝐵 ∈ V ∧ 𝐴 ∈ V ∧ 𝑓:𝐵𝐴) → 𝑓 ∈ (𝐴pm 𝐵))
62, 3, 4, 5syl3anc 1368 . 2 (𝑓 ∈ (𝐴m 𝐵) → 𝑓 ∈ (𝐴pm 𝐵))
76ssriv 3981 1 (𝐴m 𝐵) ⊆ (𝐴pm 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  Vcvv 3468  wss 3943  wf 6532  (class class class)co 7404  m cmap 8819  pm cpm 8820
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-map 8821  df-pm 8822
This theorem is referenced by:  mapsspw  8871  wunmap  10720  dvntaylp  26257  taylthlem1  26259  taylthlem2  26260  taylthlem2OLD  26261  mrsubrn  35032  mrsubff1  35033  msubrn  35048  msubff1  35075
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