Theorem mapfoss 8855

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 3461	. . . 4 ⊢ 𝑚 ∈ V
2		foeq1 6792	. . . 4 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴–onto→𝐵 ↔ 𝑚:𝐴–onto→𝐵))
3	1, 2	elab 3640	. . 3 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ↔ 𝑚:𝐴–onto→𝐵)
4		fof 6796	. . . 4 ⊢ (𝑚:𝐴–onto→𝐵 → 𝑚:𝐴⟶𝐵)
5		forn 6799	. . . . . 6 ⊢ (𝑚:𝐴–onto→𝐵 → ran 𝑚 = 𝐵)
6	1	rnex 7913	. . . . . 6 ⊢ ran 𝑚 ∈ V
7	5, 6	eqeltrrdi 2874	. . . . 5 ⊢ (𝑚:𝐴–onto→𝐵 → 𝐵 ∈ V)
8		dmfex 7908	. . . . . 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 8843	. . . 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 3942	1 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ⊆ (𝐵 ↑m 𝐴)

This proof depends on syntax axioms:   ∈ wcel 2146  {cab 2743  Vcvv 3457   ⊆ wss 3906  ran crn 5664  ⟶wf 6536  –onto→wfo 6538  (class class class)co 7419   ↑_m cmap 8830

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-map 8832

This theorem is used by:  fosetex  8861