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Theorem mapfoss 8845
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 3472 . . . 4 𝑚 ∈ V
2 foeq1 6794 . . . 4 (𝑓 = 𝑚 → (𝑓:𝐴onto𝐵𝑚:𝐴onto𝐵))
31, 2elab 3663 . . 3 (𝑚 ∈ {𝑓𝑓:𝐴onto𝐵} ↔ 𝑚:𝐴onto𝐵)
4 fof 6798 . . . 4 (𝑚:𝐴onto𝐵𝑚:𝐴𝐵)
5 forn 6801 . . . . . 6 (𝑚:𝐴onto𝐵 → ran 𝑚 = 𝐵)
61rnex 7899 . . . . . 6 ran 𝑚 ∈ V
75, 6eqeltrrdi 2836 . . . . 5 (𝑚:𝐴onto𝐵𝐵 ∈ V)
8 dmfex 7894 . . . . . 6 ((𝑚 ∈ V ∧ 𝑚:𝐴𝐵) → 𝐴 ∈ V)
91, 4, 8sylancr 586 . . . . 5 (𝑚:𝐴onto𝐵𝐴 ∈ V)
107, 9elmapd 8833 . . . 4 (𝑚:𝐴onto𝐵 → (𝑚 ∈ (𝐵m 𝐴) ↔ 𝑚:𝐴𝐵))
114, 10mpbird 257 . . 3 (𝑚:𝐴onto𝐵𝑚 ∈ (𝐵m 𝐴))
123, 11sylbi 216 . 2 (𝑚 ∈ {𝑓𝑓:𝐴onto𝐵} → 𝑚 ∈ (𝐵m 𝐴))
1312ssriv 3981 1 {𝑓𝑓:𝐴onto𝐵} ⊆ (𝐵m 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wcel 2098  {cab 2703  Vcvv 3468  wss 3943  ran crn 5670  wf 6532  ontowfo 6534  (class class class)co 7404  m cmap 8819
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fo 6542  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8821
This theorem is referenced by:  fosetex  8851
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