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Theorem elmapsnd 38905
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 4172 . . . . . . . 8 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
32fveq2d 6162 . . . . . . 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 2698 . . . . 5 ((𝜑𝑥 ∈ {𝐴}) → (𝐹𝑥) ∈ 𝐵)
87ralrimiva 2962 . . . 4 (𝜑 → ∀𝑥 ∈ {𝐴} (𝐹𝑥) ∈ 𝐵)
91, 8jca 554 . . 3 (𝜑 → (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹𝑥) ∈ 𝐵))
10 ffnfv 6354 . . 3 (𝐹:{𝐴}⟶𝐵 ↔ (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹𝑥) ∈ 𝐵))
119, 10sylibr 224 . 2 (𝜑𝐹:{𝐴}⟶𝐵)
12 elmapsnd.2 . . 3 (𝜑𝐵𝑉)
13 snex 4879 . . . 4 {𝐴} ∈ V
1413a1i 11 . . 3 (𝜑 → {𝐴} ∈ V)
1512, 14elmapd 7831 . 2 (𝜑 → (𝐹 ∈ (𝐵𝑚 {𝐴}) ↔ 𝐹:{𝐴}⟶𝐵))
1611, 15mpbird 247 1 (𝜑𝐹 ∈ (𝐵𝑚 {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wral 2908  Vcvv 3190  {csn 4155   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  𝑚 cmap 7817
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-map 7819
This theorem is referenced by:  ssmapsn  38917
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