Theorem mapfoss 8802

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 3448	. . . 4 ⊢ 𝑚 ∈ V
2		foeq1 6750	. . . 4 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴–onto→𝐵 ↔ 𝑚:𝐴–onto→𝐵))
3	1, 2	elab 3643	. . 3 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ↔ 𝑚:𝐴–onto→𝐵)
4		fof 6754	. . . 4 ⊢ (𝑚:𝐴–onto→𝐵 → 𝑚:𝐴⟶𝐵)
5		forn 6757	. . . . . 6 ⊢ (𝑚:𝐴–onto→𝐵 → ran 𝑚 = 𝐵)
6	1	rnex 7866	. . . . . 6 ⊢ ran 𝑚 ∈ V
7	5, 6	eqeltrrdi 2837	. . . . 5 ⊢ (𝑚:𝐴–onto→𝐵 → 𝐵 ∈ V)
8		dmfex 7861	. . . . . 6 ⊢ ((𝑚 ∈ V ∧ 𝑚:𝐴⟶𝐵) → 𝐴 ∈ V)
9	1, 4, 8	sylancr 587	. . . . 5 ⊢ (𝑚:𝐴–onto→𝐵 → 𝐴 ∈ V)
10	7, 9	elmapd 8790	. . . 4 ⊢ (𝑚:𝐴–onto→𝐵 → (𝑚 ∈ (𝐵 ↑m 𝐴) ↔ 𝑚:𝐴⟶𝐵))
11	4, 10	mpbird 257	. . 3 ⊢ (𝑚:𝐴–onto→𝐵 → 𝑚 ∈ (𝐵 ↑m 𝐴))
12	3, 11	sylbi 217	. 2 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} → 𝑚 ∈ (𝐵 ↑m 𝐴))
13	12	ssriv 3947	1 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ⊆ (𝐵 ↑m 𝐴)

Syntax hints:   ∈ wcel 2109  {cab 2707  Vcvv 3444   ⊆ wss 3911  ran crn 5632  ⟶wf 6495  –onto→wfo 6497  (class class class)co 7369   ↑_m cmap 8776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-fo 6505  df-fv 6507  df-ov 7372  df-oprab 7373  df-mpo 7374  df-map 8778

This theorem is referenced by:  fosetex  8808