Theorem elmapsnd 45779

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 4602	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
3	2	fveq2d 6875	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴))
4	3	adantl 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) = (𝐹‘𝐴))
5		elmapsnd.3	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝐵)
6	5	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝐴) ∈ 𝐵)
7	4, 6	eqeltrd 2865	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) ∈ 𝐵)
8	7	ralrimiva 3157	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵)
9	1, 8	jca 520	. . 3 ⊢ (𝜑 → (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵))
10		ffnfv 7104	. . 3 ⊢ (𝐹:{𝐴}⟶𝐵 ↔ (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵))
11	9, 10	sylibr 237	. 2 ⊢ (𝜑 → 𝐹:{𝐴}⟶𝐵)
12		elmapsnd.2	. . 3 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
13		snex 5401	. . . 4 ⊢ {𝐴} ∈ V
14	13	a1i 11	. . 3 ⊢ (𝜑 → {𝐴} ∈ V)
15	12, 14	elmapd 8825	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m {𝐴}) ↔ 𝐹:{𝐴}⟶𝐵))
16	11, 15	mpbird 260	1 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m {𝐴}))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  Vcvv 3457  {csn 4585   Fn wfn 6520  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ↑_m cmap 8812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-fv 6533  df-ov 7403  df-oprab 7404  df-mpo 7405  df-map 8814

This theorem is referenced by:  ssmapsn  45790