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 43784
 Description: An alternate function value belongs to the range of the function, analogous to fvelrn 6825. (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 43682 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)
2 dfatafv2rnb 43780 . 2 (𝐹 defAt 𝐴 ↔ (𝐹''''𝐴) ∈ ran 𝐹)
31, 2sylib 221 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹''''𝐴) ∈ ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2112  dom cdm 5523  ran crn 5524  Fun wfun 6322   defAt wdfat 43669  ''''cafv2 43761 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-nel 3095  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-iota 6287  df-fun 6330  df-dfat 43672  df-afv2 43762 This theorem is referenced by:  fnafv2elrn  43786
 Copyright terms: Public domain W3C validator