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Theorem mapfoss 8791
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 3444 . . . 4 𝑚 ∈ V
2 foeq1 6742 . . . 4 (𝑓 = 𝑚 → (𝑓:𝐴onto𝐵𝑚:𝐴onto𝐵))
31, 2elab 3634 . . 3 (𝑚 ∈ {𝑓𝑓:𝐴onto𝐵} ↔ 𝑚:𝐴onto𝐵)
4 fof 6746 . . . 4 (𝑚:𝐴onto𝐵𝑚:𝐴𝐵)
5 forn 6749 . . . . . 6 (𝑚:𝐴onto𝐵 → ran 𝑚 = 𝐵)
61rnex 7852 . . . . . 6 ran 𝑚 ∈ V
75, 6eqeltrrdi 2845 . . . . 5 (𝑚:𝐴onto𝐵𝐵 ∈ V)
8 dmfex 7847 . . . . . 6 ((𝑚 ∈ V ∧ 𝑚:𝐴𝐵) → 𝐴 ∈ V)
91, 4, 8sylancr 587 . . . . 5 (𝑚:𝐴onto𝐵𝐴 ∈ V)
107, 9elmapd 8779 . . . 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 3937 1 {𝑓𝑓:𝐴onto𝐵} ⊆ (𝐵m 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wcel 2113  {cab 2714  Vcvv 3440  wss 3901  ran crn 5625  wf 6488  ontowfo 6490  (class class class)co 7358  m cmap 8765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-opab 5161  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-map 8767
This theorem is referenced by:  fosetex  8797
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