Theorem dfatelrn 47730

Description: The value of a function 𝐹 at a set 𝐴 is in the range of the function 𝐹 if 𝐹 is defined at 𝐴. (Contributed by AV, 1-Sep-2022.)

Assertion

Ref	Expression
dfatelrn	⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) ∈ ran 𝐹)

Proof of Theorem dfatelrn

Step	Hyp	Ref	Expression
1		df-dfat 47718	. 2 ⊢ (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
2		funressndmfvrn 47643	. . 3 ⊢ ((Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)
3	2	ancoms 462	. 2 ⊢ ((𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})) → (𝐹‘𝐴) ∈ ran 𝐹)
4	1, 3	sylbi 219	1 ⊢ (𝐹 defAt 𝐴 → (𝐹‘𝐴) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 399   ∈ wcel 2144  {csn 4584  dom cdm 5649  ran crn 5650   ↾ cres 5651  Fun wfun 6517  ‘cfv 6523   defAt wdfat 47715

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fn 6526  df-fv 6531  df-dfat 47718

This theorem is referenced by: (None)