Theorem elmapsnd 45325

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 4592	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
3	2	fveq2d 6832	. . . . . . 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 2833	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) ∈ 𝐵)
8	7	ralrimiva 3125	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵)
9	1, 8	jca 511	. . 3 ⊢ (𝜑 → (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵))
10		ffnfv 7058	. . 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 5376	. . . 4 ⊢ {𝐴} ∈ V
14	13	a1i 11	. . 3 ⊢ (𝜑 → {𝐴} ∈ V)
15	12, 14	elmapd 8770	. 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 2113  ∀wral 3048  Vcvv 3437  {csn 4575   Fn wfn 6481  ⟶wf 6482  ‘cfv 6486  (class class class)co 7352   ↑_m cmap 8756

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674

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 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-fv 6494  df-ov 7355  df-oprab 7356  df-mpo 7357  df-map 8758

This theorem is referenced by:  ssmapsn  45337