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Theorem fvelrn 5519
Description: A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.)
Assertion
Ref Expression
fvelrn ((Fun 𝐹𝐴 ∈ 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 2180 . . . . 5 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹𝐴 ∈ dom 𝐹))
21anbi2d 459 . . . 4 (𝑥 = 𝐴 → ((Fun 𝐹𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹𝐴 ∈ dom 𝐹)))
3 fveq2 5389 . . . . 5 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
43eleq1d 2186 . . . 4 (𝑥 = 𝐴 → ((𝐹𝑥) ∈ ran 𝐹 ↔ (𝐹𝐴) ∈ ran 𝐹))
52, 4imbi12d 233 . . 3 (𝑥 = 𝐴 → (((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹) ↔ ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)))
6 funfvop 5500 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
7 vex 2663 . . . . . 6 𝑥 ∈ V
8 opeq1 3675 . . . . . . 7 (𝑦 = 𝑥 → ⟨𝑦, (𝐹𝑥)⟩ = ⟨𝑥, (𝐹𝑥)⟩)
98eleq1d 2186 . . . . . 6 (𝑦 = 𝑥 → (⟨𝑦, (𝐹𝑥)⟩ ∈ 𝐹 ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹))
107, 9spcev 2754 . . . . 5 (⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹 → ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹)
116, 10syl 14 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹)
12 funfvex 5406 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
13 elrn2g 4699 . . . . 5 ((𝐹𝑥) ∈ V → ((𝐹𝑥) ∈ ran 𝐹 ↔ ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹))
1412, 13syl 14 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) ∈ ran 𝐹 ↔ ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹))
1511, 14mpbird 166 . . 3 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
165, 15vtoclg 2720 . 2 (𝐴 ∈ dom 𝐹 → ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹))
1716anabsi7 555 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1316  wex 1453  wcel 1465  Vcvv 2660  cop 3500  dom cdm 4509  ran crn 4510  Fun wfun 5087  cfv 5093
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-sbc 2883  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-iota 5058  df-fun 5095  df-fn 5096  df-fv 5101
This theorem is referenced by:  fnfvelrn  5520  eldmrexrn  5529  funfvima  5617  elunirn  5635  frecuzrdgdomlem  10145  frecuzrdgsuctlem  10151
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