Theorem mapfoss 8447

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 3413	. . . 4 ⊢ 𝑚 ∈ V
2		foeq1 6577	. . . 4 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴–onto→𝐵 ↔ 𝑚:𝐴–onto→𝐵))
3	1, 2	elab 3590	. . 3 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ↔ 𝑚:𝐴–onto→𝐵)
4		fof 6581	. . . 4 ⊢ (𝑚:𝐴–onto→𝐵 → 𝑚:𝐴⟶𝐵)
5		forn 6584	. . . . . 6 ⊢ (𝑚:𝐴–onto→𝐵 → ran 𝑚 = 𝐵)
6	1	rnex 7628	. . . . . 6 ⊢ ran 𝑚 ∈ V
7	5, 6	eqeltrrdi 2861	. . . . 5 ⊢ (𝑚:𝐴–onto→𝐵 → 𝐵 ∈ V)
8		dmfex 7623	. . . . . 6 ⊢ ((𝑚 ∈ V ∧ 𝑚:𝐴⟶𝐵) → 𝐴 ∈ V)
9	1, 4, 8	sylancr 590	. . . . 5 ⊢ (𝑚:𝐴–onto→𝐵 → 𝐴 ∈ V)
10	7, 9	elmapd 8436	. . . 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 3898	1 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ⊆ (𝐵 ↑m 𝐴)

Syntax hints:   ∈ wcel 2111  {cab 2735  Vcvv 3409   ⊆ wss 3860  ran crn 5529  ⟶wf 6336  –onto→wfo 6338  (class class class)co 7156   ↑_m cmap 8422

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5173  ax-nul 5180  ax-pow 5238  ax-pr 5302  ax-un 7465

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-id 5434  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-iota 6299  df-fun 6342  df-fn 6343  df-f 6344  df-fo 6346  df-fv 6348  df-ov 7159  df-oprab 7160  df-mpo 7161  df-map 8424

This theorem is referenced by:  fosetex  8453