Theorem mapfoss 8888

Description: The value of the set exponentiation (𝐵 ↑_m 𝐴) is a superset of the set of all functions from 𝐴 onto 𝐵. (Contributed by AV, 7-Aug-2024.)

Assertion

Ref	Expression
mapfoss	⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ⊆ (𝐵 ↑m 𝐴)

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem mapfoss

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3458	. . . 4 ⊢ 𝑚 ∈ V
2		foeq1 6803	. . . 4 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴–onto→𝐵 ↔ 𝑚:𝐴–onto→𝐵))
3	1, 2	elab 3636	. . 3 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ↔ 𝑚:𝐴–onto→𝐵)
4		fof 6807	. . . 4 ⊢ (𝑚:𝐴–onto→𝐵 → 𝑚:𝐴⟶𝐵)
5		forn 6810	. . . . . 6 ⊢ (𝑚:𝐴–onto→𝐵 → ran 𝑚 = 𝐵)
6	1	rnex 7932	. . . . . 6 ⊢ ran 𝑚 ∈ V
7	5, 6	eqeltrrdi 2873	. . . . 5 ⊢ (𝑚:𝐴–onto→𝐵 → 𝐵 ∈ V)
8		dmfex 7927	. . . . . 6 ⊢ ((𝑚 ∈ V ∧ 𝑚:𝐴⟶𝐵) → 𝐴 ∈ V)
9	1, 4, 8	sylancr 599	. . . . 5 ⊢ (𝑚:𝐴–onto→𝐵 → 𝐴 ∈ V)
10	7, 9	elmapd 8874	. . . 4 ⊢ (𝑚:𝐴–onto→𝐵 → (𝑚 ∈ (𝐵 ↑m 𝐴) ↔ 𝑚:𝐴⟶𝐵))
11	4, 10	mpbird 260	. . 3 ⊢ (𝑚:𝐴–onto→𝐵 → 𝑚 ∈ (𝐵 ↑m 𝐴))
12	3, 11	sylbi 220	. 2 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} → 𝑚 ∈ (𝐵 ↑m 𝐴))
13	12	ssriv 3917	1 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ⊆ (𝐵 ↑m 𝐴)

This proof depends on syntax axioms:   ∈ wcel 2146  {cab 2741  Vcvv 3454   ⊆ wss 3900  ran crn 5656  ⟶wf 6543  –onto→wfo 6545  (class class class)co 7430   ↑_m cmap 8861

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2196  ax-12 2215  ax-ext 2735  ax-sep 5252  ax-pow 5331  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2101  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6501  df-fun 6549  df-fn 6550  df-f 6551  df-fo 6553  df-fv 6555  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8863

This theorem is used by:  fosetex  8894