Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  afv2elrn Structured version   Visualization version   GIF version

Theorem afv2elrn 46511
Description: An alternate function value belongs to the range of the function, analogous to fvelrn 7072. (Contributed by AV, 3-Sep-2022.)
Assertion
Ref Expression
afv2elrn ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹)

Proof of Theorem afv2elrn
StepHypRef Expression
1 fundmdfat 46409 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)
2 dfatafv2rnb 46507 . 2 (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
31, 2sylib 217 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2098  dom cdm 5669  ran crn 5670  Fun wfun 6531   defAt wdfat 46396  ''''cafv2 46488
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-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  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-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  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-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6489  df-fun 6539  df-dfat 46399  df-afv2 46489
This theorem is referenced by:  fnafv2elrn  46513
  Copyright terms: Public domain W3C validator