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Theorem fvelrn 5241
Description: A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
Assertion
Ref Expression
fvelrn ((Fun 𝐹 A dom 𝐹) → (𝐹A) ran 𝐹)

Proof of Theorem fvelrn
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2097 . . . . 5 (x = A → (x dom 𝐹A dom 𝐹))
21anbi2d 437 . . . 4 (x = A → ((Fun 𝐹 x dom 𝐹) ↔ (Fun 𝐹 A dom 𝐹)))
3 fveq2 5121 . . . . 5 (x = A → (𝐹x) = (𝐹A))
43eleq1d 2103 . . . 4 (x = A → ((𝐹x) ran 𝐹 ↔ (𝐹A) ran 𝐹))
52, 4imbi12d 223 . . 3 (x = A → (((Fun 𝐹 x dom 𝐹) → (𝐹x) ran 𝐹) ↔ ((Fun 𝐹 A dom 𝐹) → (𝐹A) ran 𝐹)))
6 funfvop 5222 . . . . 5 ((Fun 𝐹 x dom 𝐹) → ⟨x, (𝐹x)⟩ 𝐹)
7 vex 2554 . . . . . 6 x V
8 opeq1 3540 . . . . . . 7 (y = x → ⟨y, (𝐹x)⟩ = ⟨x, (𝐹x)⟩)
98eleq1d 2103 . . . . . 6 (y = x → (⟨y, (𝐹x)⟩ 𝐹 ↔ ⟨x, (𝐹x)⟩ 𝐹))
107, 9spcev 2641 . . . . 5 (⟨x, (𝐹x)⟩ 𝐹yy, (𝐹x)⟩ 𝐹)
116, 10syl 14 . . . 4 ((Fun 𝐹 x dom 𝐹) → yy, (𝐹x)⟩ 𝐹)
12 funfvex 5135 . . . . 5 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
13 elrn2g 4468 . . . . 5 ((𝐹x) V → ((𝐹x) ran 𝐹yy, (𝐹x)⟩ 𝐹))
1412, 13syl 14 . . . 4 ((Fun 𝐹 x dom 𝐹) → ((𝐹x) ran 𝐹yy, (𝐹x)⟩ 𝐹))
1511, 14mpbird 156 . . 3 ((Fun 𝐹 x dom 𝐹) → (𝐹x) ran 𝐹)
165, 15vtoclg 2607 . 2 (A dom 𝐹 → ((Fun 𝐹 A dom 𝐹) → (𝐹A) ran 𝐹))
1716anabsi7 515 1 ((Fun 𝐹 A dom 𝐹) → (𝐹A) ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  cop 3370  dom cdm 4288  ran crn 4289  Fun wfun 4839  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-bnd 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:  fnfvelrn  5242  eldmrexrn  5251  funfvima  5333  elunirn  5348
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