Theorem elmapsnd 45147

Description: Membership in a set exponentiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
elmapsnd.1	⊢ (𝜑 → 𝐹 Fn {𝐴})
elmapsnd.2	⊢ (𝜑 → 𝐵 ∈ 𝑉)
elmapsnd.3	⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝐵)

Assertion

Ref	Expression
elmapsnd	⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m {𝐴}))

Proof of Theorem elmapsnd

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elmapsnd.1	. . . 4 ⊢ (𝜑 → 𝐹 Fn {𝐴})
2		elsni 4648	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
3	2	fveq2d 6911	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴))
4	3	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) = (𝐹‘𝐴))
5		elmapsnd.3	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝐵)
6	5	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝐴) ∈ 𝐵)
7	4, 6	eqeltrd 2839	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) ∈ 𝐵)
8	7	ralrimiva 3144	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵)
9	1, 8	jca 511	. . 3 ⊢ (𝜑 → (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵))
10		ffnfv 7139	. . 3 ⊢ (𝐹:{𝐴}⟶𝐵 ↔ (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵))
11	9, 10	sylibr 234	. 2 ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵)
12		elmapsnd.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
13		snex 5442	. . . 4 ⊢ {𝐴} ∈ V
14	13	a1i 11	. . 3 ⊢ (𝜑 → {𝐴} ∈ V)
15	12, 14	elmapd 8879	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m {𝐴}) ↔ 𝐹:{𝐴}⟶𝐵))
16	11, 15	mpbird 257	1 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m {𝐴}))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  Vcvv 3478  {csn 4631   Fn wfn 6558  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431   ↑_m cmap 8865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-map 8867

This theorem is referenced by:  ssmapsn  45159