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Theorem mapfoss 8875
Description: The value of the set exponentiation (𝐵m 𝐴) is a superset of the set of all functions from 𝐴 onto 𝐵. (Contributed by AV, 7-Aug-2024.)
Assertion
Ref Expression
mapfoss {𝑓𝑓:𝐴onto𝐵} ⊆ (𝐵m 𝐴)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem mapfoss
Dummy variable 𝑚 is distinct from all other variables.
StepHypRef Expression
1 vex 3475 . . . 4 𝑚 ∈ V
2 foeq1 6810 . . . 4 (𝑓 = 𝑚 → (𝑓:𝐴onto𝐵𝑚:𝐴onto𝐵))
31, 2elab 3667 . . 3 (𝑚 ∈ {𝑓𝑓:𝐴onto𝐵} ↔ 𝑚:𝐴onto𝐵)
4 fof 6814 . . . 4 (𝑚:𝐴onto𝐵𝑚:𝐴𝐵)
5 forn 6817 . . . . . 6 (𝑚:𝐴onto𝐵 → ran 𝑚 = 𝐵)
61rnex 7922 . . . . . 6 ran 𝑚 ∈ V
75, 6eqeltrrdi 2837 . . . . 5 (𝑚:𝐴onto𝐵𝐵 ∈ V)
8 dmfex 7917 . . . . . 6 ((𝑚 ∈ V ∧ 𝑚:𝐴𝐵) → 𝐴 ∈ V)
91, 4, 8sylancr 585 . . . . 5 (𝑚:𝐴onto𝐵𝐴 ∈ V)
107, 9elmapd 8863 . . . 4 (𝑚:𝐴onto𝐵 → (𝑚 ∈ (𝐵m 𝐴) ↔ 𝑚:𝐴𝐵))
114, 10mpbird 256 . . 3 (𝑚:𝐴onto𝐵𝑚 ∈ (𝐵m 𝐴))
123, 11sylbi 216 . 2 (𝑚 ∈ {𝑓𝑓:𝐴onto𝐵} → 𝑚 ∈ (𝐵m 𝐴))
1312ssriv 3984 1 {𝑓𝑓:𝐴onto𝐵} ⊆ (𝐵m 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  {cab 2704  Vcvv 3471  wss 3947  ran crn 5681  wf 6547  ontowfo 6549  (class class class)co 7424  m cmap 8849
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-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  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-br 5151  df-opab 5213  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-fo 6557  df-fv 6559  df-ov 7427  df-oprab 7428  df-mpo 7429  df-map 8851
This theorem is referenced by:  fosetex  8881
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