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Theorem mapfoss 8906
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 3486 . . . 4 𝑚 ∈ V
2 foeq1 6829 . . . 4 (𝑓 = 𝑚 → (𝑓:𝐴onto𝐵𝑚:𝐴onto𝐵))
31, 2elab 3689 . . 3 (𝑚 ∈ {𝑓𝑓:𝐴onto𝐵} ↔ 𝑚:𝐴onto𝐵)
4 fof 6833 . . . 4 (𝑚:𝐴onto𝐵𝑚:𝐴𝐵)
5 forn 6836 . . . . . 6 (𝑚:𝐴onto𝐵 → ran 𝑚 = 𝐵)
61rnex 7946 . . . . . 6 ran 𝑚 ∈ V
75, 6eqeltrrdi 2847 . . . . 5 (𝑚:𝐴onto𝐵𝐵 ∈ V)
8 dmfex 7941 . . . . . 6 ((𝑚 ∈ V ∧ 𝑚:𝐴𝐵) → 𝐴 ∈ V)
91, 4, 8sylancr 586 . . . . 5 (𝑚:𝐴onto𝐵𝐴 ∈ V)
107, 9elmapd 8894 . . . 4 (𝑚:𝐴onto𝐵 → (𝑚 ∈ (𝐵m 𝐴) ↔ 𝑚:𝐴𝐵))
114, 10mpbird 257 . . 3 (𝑚:𝐴onto𝐵𝑚 ∈ (𝐵m 𝐴))
123, 11sylbi 217 . 2 (𝑚 ∈ {𝑓𝑓:𝐴onto𝐵} → 𝑚 ∈ (𝐵m 𝐴))
1312ssriv 4006 1 {𝑓𝑓:𝐴onto𝐵} ⊆ (𝐵m 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wcel 2103  {cab 2711  Vcvv 3482  wss 3970  ran crn 5700  wf 6568  ontowfo 6570  (class class class)co 7445  m cmap 8880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pow 5386  ax-pr 5450  ax-un 7766
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2890  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-sbc 3799  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5170  df-opab 5232  df-id 5597  df-xp 5705  df-rel 5706  df-cnv 5707  df-co 5708  df-dm 5709  df-rn 5710  df-iota 6524  df-fun 6574  df-fn 6575  df-f 6576  df-fo 6578  df-fv 6580  df-ov 7448  df-oprab 7449  df-mpo 7450  df-map 8882
This theorem is referenced by:  fosetex  8912
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