Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfatsnafv2 Structured version   Visualization version   GIF version

Theorem dfatsnafv2 42106
Description: Singleton of function value, analogous to fnsnfv 6483. (Contributed by AV, 7-Sep-2022.)
Assertion
Ref Expression
dfatsnafv2 (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴}))

Proof of Theorem dfatsnafv2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqcom 2806 . . . 4 (𝑦 = (𝐹''''𝐴) ↔ (𝐹''''𝐴) = 𝑦)
2 dfatbrafv2b 42099 . . . . 5 ((𝐹 defAt 𝐴𝑦 ∈ V) → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
32elvd 3390 . . . 4 (𝐹 defAt 𝐴 → ((𝐹''''𝐴) = 𝑦𝐴𝐹𝑦))
41, 3syl5bb 275 . . 3 (𝐹 defAt 𝐴 → (𝑦 = (𝐹''''𝐴) ↔ 𝐴𝐹𝑦))
54abbidv 2918 . 2 (𝐹 defAt 𝐴 → {𝑦𝑦 = (𝐹''''𝐴)} = {𝑦𝐴𝐹𝑦})
6 df-sn 4369 . . 3 {(𝐹''''𝐴)} = {𝑦𝑦 = (𝐹''''𝐴)}
76a1i 11 . 2 (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = {𝑦𝑦 = (𝐹''''𝐴)})
8 dfdfat2 41982 . . 3 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥))
9 imasng 5704 . . . 4 (𝐴 ∈ dom 𝐹 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
109adantr 473 . . 3 ((𝐴 ∈ dom 𝐹 ∧ ∃!𝑥 𝐴𝐹𝑥) → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
118, 10sylbi 209 . 2 (𝐹 defAt 𝐴 → (𝐹 “ {𝐴}) = {𝑦𝐴𝐹𝑦})
125, 7, 113eqtr4d 2843 1 (𝐹 defAt 𝐴 → {(𝐹''''𝐴)} = (𝐹 “ {𝐴}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  ∃!weu 2608  {cab 2785  Vcvv 3385  {csn 4368   class class class wbr 4843  dom cdm 5312  cima 5315   defAt wdfat 41970  ''''cafv2 42062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-dfat 41973  df-afv2 42063
This theorem is referenced by:  afv2co2  42111
  Copyright terms: Public domain W3C validator