Theorem fnfvelrn 5242

Description: A function's value belongs to its range. (Contributed by NM, 15-Oct-1996.)

Assertion

Ref	Expression
fnfvelrn	⊢ ((𝐹 Fn A ∧ B ∈ A) → (𝐹‘B) ∈ ran 𝐹)

Proof of Theorem fnfvelrn

Step	Hyp	Ref	Expression
1		fvelrn 5241	. 2 ⊢ ((Fun 𝐹 ∧ B ∈ dom 𝐹) → (𝐹‘B) ∈ ran 𝐹)
2	1	funfni 4942	1 ⊢ ((𝐹 Fn A ∧ B ∈ A) → (𝐹‘B) ∈ ran 𝐹)

Syntax hints:   → wi 4   ∧ wa 97   ∈ wcel 1390  ran crn 4289   Fn wfn 4840  ‘cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by:  ffvelrn  5243  fnovrn  5590  fo1stresm  5730  fo2ndresm  5731  fo2ndf  5790  frec2uzrand  8872  frecuzrdglem  8878  frecuzrdg0  8881