Theorem elmapsnd 46423

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 4601	. . . . . . . 8 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
3	2	fveq2d 6912	. . . . . . 7 ⊢ (𝑥 ∈ {𝐴} → (𝐹‘𝑥) = (𝐹‘𝐴))
4	3	adantl 487	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) = (𝐹‘𝐴))
5		elmapsnd.3	. . . . . . 7 ⊢ (𝜑 → (𝐹‘𝐴) ∈ 𝐵)
6	5	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝐴) ∈ 𝐵)
7	4, 6	eqeltrd 2864	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → (𝐹‘𝑥) ∈ 𝐵)
8	7	ralrimiva 3158	. . . 4 ⊢ (𝜑 → ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵)
9	1, 8	jca 521	. . 3 ⊢ (𝜑 → (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹‘𝑥) ∈ 𝐵))
10		ffnfv 7143	. . 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 8888	. 2 ⊢ (𝜑 → (𝐹 ∈ (𝐵 ↑m {𝐴}) ↔ 𝐹:{𝐴}⟶𝐵))
16	11, 15	mpbird 260	1 ⊢ (𝜑 → 𝐹 ∈ (𝐵 ↑m {𝐴}))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454  {csn 4588   Fn wfn 6553  ⟶wf 6554  ‘cfv 6558  (class class class)co 7443   ↑_m cmap 8875

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7775

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-mpt 5190  df-id 5550  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-iota 6512  df-fun 6560  df-fn 6561  df-f 6562  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448  df-map 8877

This theorem is used by:  ssmapsn  46434