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Theorem elmapsnd 41758
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 4567 . . . . . . . 8 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
32fveq2d 6665 . . . . . . 7 (𝑥 ∈ {𝐴} → (𝐹𝑥) = (𝐹𝐴))
43adantl 485 . . . . . 6 ((𝜑𝑥 ∈ {𝐴}) → (𝐹𝑥) = (𝐹𝐴))
5 elmapsnd.3 . . . . . . 7 (𝜑 → (𝐹𝐴) ∈ 𝐵)
65adantr 484 . . . . . 6 ((𝜑𝑥 ∈ {𝐴}) → (𝐹𝐴) ∈ 𝐵)
74, 6eqeltrd 2916 . . . . 5 ((𝜑𝑥 ∈ {𝐴}) → (𝐹𝑥) ∈ 𝐵)
87ralrimiva 3177 . . . 4 (𝜑 → ∀𝑥 ∈ {𝐴} (𝐹𝑥) ∈ 𝐵)
91, 8jca 515 . . 3 (𝜑 → (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹𝑥) ∈ 𝐵))
10 ffnfv 6873 . . 3 (𝐹:{𝐴}⟶𝐵 ↔ (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹𝑥) ∈ 𝐵))
119, 10sylibr 237 . 2 (𝜑𝐹:{𝐴}⟶𝐵)
12 elmapsnd.2 . . 3 (𝜑𝐵𝑉)
13 snex 5319 . . . 4 {𝐴} ∈ V
1413a1i 11 . . 3 (𝜑 → {𝐴} ∈ V)
1512, 14elmapd 8416 . 2 (𝜑 → (𝐹 ∈ (𝐵m {𝐴}) ↔ 𝐹:{𝐴}⟶𝐵))
1611, 15mpbird 260 1 (𝜑𝐹 ∈ (𝐵m {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  wral 3133  Vcvv 3480  {csn 4550   Fn wfn 6338  wf 6339  cfv 6343  (class class class)co 7149  m cmap 8402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-fv 6351  df-ov 7152  df-oprab 7153  df-mpo 7154  df-map 8404
This theorem is referenced by:  ssmapsn  41770
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