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Theorem elmapsnd 41343
Description: Membership in a set exponentiated to a singleton. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
elmapsnd.1 (𝜑𝐹 Fn {𝐴})
elmapsnd.2 (𝜑𝐵𝑉)
elmapsnd.3 (𝜑 → (𝐹𝐴) ∈ 𝐵)
Assertion
Ref Expression
elmapsnd (𝜑𝐹 ∈ (𝐵m {𝐴}))

Proof of Theorem elmapsnd
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elmapsnd.1 . . . 4 (𝜑𝐹 Fn {𝐴})
2 elsni 4574 . . . . . . . 8 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
32fveq2d 6667 . . . . . . 7 (𝑥 ∈ {𝐴} → (𝐹𝑥) = (𝐹𝐴))
43adantl 482 . . . . . 6 ((𝜑𝑥 ∈ {𝐴}) → (𝐹𝑥) = (𝐹𝐴))
5 elmapsnd.3 . . . . . . 7 (𝜑 → (𝐹𝐴) ∈ 𝐵)
65adantr 481 . . . . . 6 ((𝜑𝑥 ∈ {𝐴}) → (𝐹𝐴) ∈ 𝐵)
74, 6eqeltrd 2910 . . . . 5 ((𝜑𝑥 ∈ {𝐴}) → (𝐹𝑥) ∈ 𝐵)
87ralrimiva 3179 . . . 4 (𝜑 → ∀𝑥 ∈ {𝐴} (𝐹𝑥) ∈ 𝐵)
91, 8jca 512 . . 3 (𝜑 → (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹𝑥) ∈ 𝐵))
10 ffnfv 6874 . . 3 (𝐹:{𝐴}⟶𝐵 ↔ (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹𝑥) ∈ 𝐵))
119, 10sylibr 235 . 2 (𝜑𝐹:{𝐴}⟶𝐵)
12 elmapsnd.2 . . 3 (𝜑𝐵𝑉)
13 snex 5322 . . . 4 {𝐴} ∈ V
1413a1i 11 . . 3 (𝜑 → {𝐴} ∈ V)
1512, 14elmapd 8409 . 2 (𝜑 → (𝐹 ∈ (𝐵m {𝐴}) ↔ 𝐹:{𝐴}⟶𝐵))
1611, 15mpbird 258 1 (𝜑𝐹 ∈ (𝐵m {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105  wral 3135  Vcvv 3492  {csn 4557   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  m cmap 8395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-map 8397
This theorem is referenced by:  ssmapsn  41355
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