Theorem mapfoss 8834

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 3459	. . . 4 ⊢ 𝑚 ∈ V
2		foeq1 6775	. . . 4 ⊢ (𝑓 = 𝑚 → (𝑓:𝐴–onto→𝐵 ↔ 𝑚:𝐴–onto→𝐵))
3	1, 2	elab 3639	. . 3 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ↔ 𝑚:𝐴–onto→𝐵)
4		fof 6779	. . . 4 ⊢ (𝑚:𝐴–onto→𝐵 → 𝑚:𝐴⟶𝐵)
5		forn 6782	. . . . . 6 ⊢ (𝑚:𝐴–onto→𝐵 → ran 𝑚 = 𝐵)
6	1	rnex 7892	. . . . . 6 ⊢ ran 𝑚 ∈ V
7	5, 6	eqeltrrdi 2872	. . . . 5 ⊢ (𝑚:𝐴–onto→𝐵 → 𝐵 ∈ V)
8		dmfex 7887	. . . . . 6 ⊢ ((𝑚 ∈ V ∧ 𝑚:𝐴⟶𝐵) → 𝐴 ∈ V)
9	1, 4, 8	sylancr 596	. . . . 5 ⊢ (𝑚:𝐴–onto→𝐵 → 𝐴 ∈ V)
10	7, 9	elmapd 8822	. . . 4 ⊢ (𝑚:𝐴–onto→𝐵 → (𝑚 ∈ (𝐵 ↑m 𝐴) ↔ 𝑚:𝐴⟶𝐵))
11	4, 10	mpbird 259	. . 3 ⊢ (𝑚:𝐴–onto→𝐵 → 𝑚 ∈ (𝐵 ↑m 𝐴))
12	3, 11	sylbi 219	. 2 ⊢ (𝑚 ∈ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} → 𝑚 ∈ (𝐵 ↑m 𝐴))
13	12	ssriv 3941	1 ⊢ {𝑓 ∣ 𝑓:𝐴–onto→𝐵} ⊆ (𝐵 ↑m 𝐴)

Syntax hints:   ∈ wcel 2143  {cab 2741  Vcvv 3455   ⊆ wss 3905  ran crn 5649  ⟶wf 6518  –onto→wfo 6520  (class class class)co 7397   ↑_m cmap 8809

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-pow 5323  ax-pr 5391  ax-un 7719

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-nf 1805  df-sb 2092  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-sbc 3746  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-opab 5164  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-iota 6478  df-fun 6524  df-fn 6525  df-f 6526  df-fo 6528  df-fv 6530  df-ov 7400  df-oprab 7401  df-mpo 7402  df-map 8811

This theorem is referenced by:  fosetex  8840