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Theorem afvelrn 46174
Description: A function's value belongs to its range, analogous to fvelrn 7077. (Contributed by Alexander van der Vekens, 25-May-2017.)
Assertion
Ref Expression
afvelrn ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹)

Proof of Theorem afvelrn
StepHypRef Expression
1 funres 6589 . . . . . 6 (Fun 𝐹 → Fun (𝐹 ↾ {𝐴}))
21anim1i 613 . . . . 5 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (Fun (𝐹 ↾ {𝐴}) ∧ 𝐴 ∈ dom 𝐹))
32ancomd 460 . . . 4 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
4 df-dfat 46125 . . . 4 (𝐹 defAt 𝐴 ↔ (𝐴 ∈ dom 𝐹 ∧ Fun (𝐹 ↾ {𝐴})))
53, 4sylibr 233 . . 3 ((Fun 𝐹𝐴 ∈ dom 𝐹) → 𝐹 defAt 𝐴)
6 afvfundmfveq 46144 . . . 4 (𝐹 defAt 𝐴 → (𝐹'''𝐴) = (𝐹𝐴))
76eqcomd 2736 . . 3 (𝐹 defAt 𝐴 → (𝐹𝐴) = (𝐹'''𝐴))
85, 7syl 17 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) = (𝐹'''𝐴))
9 fvelrn 7077 . 2 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
108, 9eqeltrrd 2832 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹'''𝐴) ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1539  wcel 2104  {csn 4627  dom cdm 5675  ran crn 5676  cres 5677  Fun wfun 6536  cfv 6542   defAt wdfat 46122  '''cafv 46123
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-aiota 46091  df-dfat 46125  df-afv 46126
This theorem is referenced by:  fnafvelrn  46175
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