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Theorem elmapsnd 44475
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 4640 . . . . . . . 8 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
32fveq2d 6889 . . . . . . 7 (𝑥 ∈ {𝐴} → (𝐹𝑥) = (𝐹𝐴))
43adantl 481 . . . . . 6 ((𝜑𝑥 ∈ {𝐴}) → (𝐹𝑥) = (𝐹𝐴))
5 elmapsnd.3 . . . . . . 7 (𝜑 → (𝐹𝐴) ∈ 𝐵)
65adantr 480 . . . . . 6 ((𝜑𝑥 ∈ {𝐴}) → (𝐹𝐴) ∈ 𝐵)
74, 6eqeltrd 2827 . . . . 5 ((𝜑𝑥 ∈ {𝐴}) → (𝐹𝑥) ∈ 𝐵)
87ralrimiva 3140 . . . 4 (𝜑 → ∀𝑥 ∈ {𝐴} (𝐹𝑥) ∈ 𝐵)
91, 8jca 511 . . 3 (𝜑 → (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹𝑥) ∈ 𝐵))
10 ffnfv 7114 . . 3 (𝐹:{𝐴}⟶𝐵 ↔ (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹𝑥) ∈ 𝐵))
119, 10sylibr 233 . 2 (𝜑𝐹:{𝐴}⟶𝐵)
12 elmapsnd.2 . . 3 (𝜑𝐵𝑉)
13 snex 5424 . . . 4 {𝐴} ∈ V
1413a1i 11 . . 3 (𝜑 → {𝐴} ∈ V)
1512, 14elmapd 8836 . 2 (𝜑 → (𝐹 ∈ (𝐵m {𝐴}) ↔ 𝐹:{𝐴}⟶𝐵))
1611, 15mpbird 257 1 (𝜑𝐹 ∈ (𝐵m {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3055  Vcvv 3468  {csn 4623   Fn wfn 6532  wf 6533  cfv 6537  (class class class)co 7405  m cmap 8822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-map 8824
This theorem is referenced by:  ssmapsn  44487
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