Theorem elmapsnd 45240

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 4593	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
3	2	fveq2d 6826	. . . . . . 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 2831	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) ∈ 𝐵)
8	7	ralrimiva 3124	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵)
9	1, 8	jca 511	. . 3 ⊢ (𝜑 → (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵))
10		ffnfv 7052	. . 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 5374	. . . 4 ⊢ {𝐴} ∈ V
14	13	a1i 11	. . 3 ⊢ (𝜑 → {𝐴} ∈ V)
15	12, 14	elmapd 8764	. 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 1541   ∈ wcel 2111  ∀wral 3047  Vcvv 3436  {csn 4576   Fn wfn 6476  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346   ↑_m cmap 8750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pow 5303  ax-pr 5370  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-map 8752

This theorem is referenced by:  ssmapsn  45252