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Theorem mapsspm 6660
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 6647 . . . 4 (𝑓 ∈ (𝐴𝑚 𝐵) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
21simprd 113 . . 3 (𝑓 ∈ (𝐴𝑚 𝐵) → 𝐵 ∈ V)
31simpld 111 . . 3 (𝑓 ∈ (𝐴𝑚 𝐵) → 𝐴 ∈ V)
4 elmapi 6648 . . 3 (𝑓 ∈ (𝐴𝑚 𝐵) → 𝑓:𝐵𝐴)
5 fpmg 6652 . . 3 ((𝐵 ∈ V ∧ 𝐴 ∈ V ∧ 𝑓:𝐵𝐴) → 𝑓 ∈ (𝐴pm 𝐵))
62, 3, 4, 5syl3anc 1233 . 2 (𝑓 ∈ (𝐴𝑚 𝐵) → 𝑓 ∈ (𝐴pm 𝐵))
76ssriv 3151 1 (𝐴𝑚 𝐵) ⊆ (𝐴pm 𝐵)
Colors of variables: wff set class
Syntax hints:  wcel 2141  Vcvv 2730  wss 3121  wf 5194  (class class class)co 5853  𝑚 cmap 6626  pm cpm 6627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-map 6628  df-pm 6629
This theorem is referenced by:  mapsspw  6662
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