Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elmapsnd Structured version   Visualization version   GIF version

Theorem elmapsnd 40317
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 4415 . . . . . . . 8 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
32fveq2d 6450 . . . . . . 7 (𝑥 ∈ {𝐴} → (𝐹𝑥) = (𝐹𝐴))
43adantl 475 . . . . . 6 ((𝜑𝑥 ∈ {𝐴}) → (𝐹𝑥) = (𝐹𝐴))
5 elmapsnd.3 . . . . . . 7 (𝜑 → (𝐹𝐴) ∈ 𝐵)
65adantr 474 . . . . . 6 ((𝜑𝑥 ∈ {𝐴}) → (𝐹𝐴) ∈ 𝐵)
74, 6eqeltrd 2859 . . . . 5 ((𝜑𝑥 ∈ {𝐴}) → (𝐹𝑥) ∈ 𝐵)
87ralrimiva 3148 . . . 4 (𝜑 → ∀𝑥 ∈ {𝐴} (𝐹𝑥) ∈ 𝐵)
91, 8jca 507 . . 3 (𝜑 → (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹𝑥) ∈ 𝐵))
10 ffnfv 6652 . . 3 (𝐹:{𝐴}⟶𝐵 ↔ (𝐹 Fn {𝐴} ∧ ∀𝑥 ∈ {𝐴} (𝐹𝑥) ∈ 𝐵))
119, 10sylibr 226 . 2 (𝜑𝐹:{𝐴}⟶𝐵)
12 elmapsnd.2 . . 3 (𝜑𝐵𝑉)
13 snex 5140 . . . 4 {𝐴} ∈ V
1413a1i 11 . . 3 (𝜑 → {𝐴} ∈ V)
1512, 14elmapd 8154 . 2 (𝜑 → (𝐹 ∈ (𝐵𝑚 {𝐴}) ↔ 𝐹:{𝐴}⟶𝐵))
1611, 15mpbird 249 1 (𝜑𝐹 ∈ (𝐵𝑚 {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386   = wceq 1601  wcel 2107  wral 3090  Vcvv 3398  {csn 4398   Fn wfn 6130  wf 6131  cfv 6135  (class class class)co 6922  𝑚 cmap 8140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-sbc 3653  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-map 8142
This theorem is referenced by:  ssmapsn  40329
  Copyright terms: Public domain W3C validator