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Theorem mapfoss 8790
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 3449 . . . 4 𝑚 ∈ V
2 foeq1 6752 . . . 4 (𝑓 = 𝑚 → (𝑓:𝐴onto𝐵𝑚:𝐴onto𝐵))
31, 2elab 3630 . . 3 (𝑚 ∈ {𝑓𝑓:𝐴onto𝐵} ↔ 𝑚:𝐴onto𝐵)
4 fof 6756 . . . 4 (𝑚:𝐴onto𝐵𝑚:𝐴𝐵)
5 forn 6759 . . . . . 6 (𝑚:𝐴onto𝐵 → ran 𝑚 = 𝐵)
61rnex 7849 . . . . . 6 ran 𝑚 ∈ V
75, 6eqeltrrdi 2847 . . . . 5 (𝑚:𝐴onto𝐵𝐵 ∈ V)
8 dmfex 7844 . . . . . 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 256 . . 3 (𝑚:𝐴onto𝐵𝑚 ∈ (𝐵m 𝐴))
123, 11sylbi 216 . 2 (𝑚 ∈ {𝑓𝑓:𝐴onto𝐵} → 𝑚 ∈ (𝐵m 𝐴))
1312ssriv 3948 1 {𝑓𝑓:𝐴onto𝐵} ⊆ (𝐵m 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wcel 2106  {cab 2713  Vcvv 3445  wss 3910  ran crn 5634  wf 6492  ontowfo 6494  (class class class)co 7357  m cmap 8765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-fo 6502  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-map 8767
This theorem is referenced by:  fosetex  8796
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