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Theorem fvelrn 5627
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 2233 . . . . 5 (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹𝐴 ∈ dom 𝐹))
21anbi2d 461 . . . 4 (𝑥 = 𝐴 → ((Fun 𝐹𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹𝐴 ∈ dom 𝐹)))
3 fveq2 5496 . . . . 5 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
43eleq1d 2239 . . . 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 5608 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹)
7 vex 2733 . . . . . 6 𝑥 ∈ V
8 opeq1 3765 . . . . . . 7 (𝑦 = 𝑥 → ⟨𝑦, (𝐹𝑥)⟩ = ⟨𝑥, (𝐹𝑥)⟩)
98eleq1d 2239 . . . . . 6 (𝑦 = 𝑥 → (⟨𝑦, (𝐹𝑥)⟩ ∈ 𝐹 ↔ ⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹))
107, 9spcev 2825 . . . . 5 (⟨𝑥, (𝐹𝑥)⟩ ∈ 𝐹 → ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹)
116, 10syl 14 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ∃𝑦𝑦, (𝐹𝑥)⟩ ∈ 𝐹)
12 funfvex 5513 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
13 elrn2g 4801 . . . . 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 2790 . 2 (𝐴 ∈ dom 𝐹 → ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹))
1716anabsi7 576 1 ((Fun 𝐹𝐴 ∈ dom 𝐹) → (𝐹𝐴) ∈ ran 𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1348  wex 1485  wcel 2141  Vcvv 2730  cop 3586  dom cdm 4611  ran crn 4612  Fun wfun 5192  cfv 5198
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-iota 5160  df-fun 5200  df-fn 5201  df-fv 5206
This theorem is referenced by:  fnfvelrn  5628  eldmrexrn  5637  funfvima  5727  elunirn  5745  frecuzrdgdomlem  10373  frecuzrdgsuctlem  10379
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